8.2.2, we get that to show $(\gamma[\mu]\otimes\nu)(B^{\prime\prime})=1$ it
suffices to show that for every $(x,v)\in R$ and for the set

$\displaystyle U(x,v)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\left\\{(x^{\prime},y,z)\in\phi^{-1}(x,v)\;\middle|\;G^{\prime}\Bigl{(}x,v,\rho(x,v)\bigl{(}L(x,y,z)\bigr{)}\Bigr{)}<y\right\\},$

we have $\rho(x,v)(U(x,v))=0$.

Fix $(x,v)\in R$ and since we will be working with points of the form
$(x^{\prime},y,z)\in\phi^{-1}(x,v)$, which forces $x^{\prime}=x$, we will
simplify notation by writing $(x,y,z)\in\phi^{-1}(x,v)$. We will also use the
shorthand $U\stackrel{{\scriptstyle\text{def}}}{{=}}U(x,v)$.

Let us define by transfinite induction sequences
$(y_{\alpha})_{\alpha<\alpha_{0}}$,
$(y^{\prime}_{\alpha})_{\alpha<\alpha_{0}}$,
$(z_{\alpha})_{\alpha<\alpha_{0}}$ and $(M_{\alpha})_{\alpha<\alpha_{0}}$ in
$[0,1]$ with the following properties:

  1. a.

$y_{\alpha}<y^{\prime}_{\alpha}$.

  2. b.

If $(x,y^{\prime}_{\alpha},z_{\alpha})\in U$, then
$(x,y^{\prime}_{\alpha},z_{\alpha})$ is _not_ an atom of the measure
$\rho(x,v)$, that is, we have
$\rho(x,v)(\\{(x,y^{\prime}_{\alpha},z_{\alpha})\\})=0$.

  3. c.

For the set

$\displaystyle U_{\alpha}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\\{(x,y,z)\in
U\mid(y_{\alpha},1)<_{L}(y,z)\leq_{L}(y^{\prime}_{\alpha},z_{\alpha})\\},$
(8.5)

we have $\rho(x,v)(L(x,y,z))=M_{\alpha}$ for every $(x,y,z)\in U_{\alpha}$.

  4. d.

$\rho(x,v)(U_{\alpha})=0$ for $U_{\alpha}$ given by (8.5).

  5. e.

For every $\beta<\alpha$, we have
$(y_{\beta},y^{\prime}_{\beta})\not\supseteq(y_{\alpha},y^{\prime}_{\alpha})$
(as open intervals in $[0,1]$).

Given $(y_{\beta})_{\beta<\alpha}$, $(y^{\prime}_{\beta})_{\beta<\alpha}$ and
$(z_{\beta})_{\beta<\alpha}$, if $U\subseteq\bigcup_{\beta<\alpha}U_{\beta}$,
then stop the construction setting
$\alpha_{0}\stackrel{{\scriptstyle\text{def}}}{{=}}\alpha$; otherwise, let
$(x,y^{\prime}_{\alpha},\widetilde{z}_{\alpha})\in
U\setminus\bigcup_{\beta<\alpha}U_{\beta}$ and define

$\displaystyle M_{\alpha}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\rho(x,v)\bigl{(}L(x,y^{\prime}_{\alpha},\widetilde{z}_{\alpha})\bigr{)},$
$\displaystyle y_{\alpha}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}G^{\prime}(x,v,M_{\alpha}),$
$\displaystyle z_{\alpha}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\sup\left\\{z\in
Z\;\middle|\;\rho(x,v)\bigl{(}L(x,y^{\prime}_{\alpha},z)\bigr{)}=M_{\alpha}\right\\}.$

(Note that in the definition of $z_{\alpha}$, all points $z$ in the set
satisfy $\phi(x,y^{\prime}_{\alpha},z)=(x,v)$ since
$(x,y^{\prime}_{\alpha},\widetilde{z}_{\alpha})\in U$.)

Let us check that all promised items are inductively satisfied.

First note that since $(x,y^{\prime}_{\alpha},\widetilde{z}_{\alpha})\in U$,
we must have

$\displaystyle y_{\alpha}$
$\displaystyle=G^{\prime}\Bigl{(}x,v,\rho(x,v)\bigl{(}L(x,y,z)\bigr{)}\Bigr{)}<y^{\prime}_{\alpha},$

so item (a) holds.

We prove item (b) by the contra-positive. Suppose that
$(x,y^{\prime}_{\alpha},z_{\alpha})$ is an atom of $\rho(x,v)$ and note that
the definition of $G^{\prime}$ in (8.3) implies that

$\displaystyle
G^{\prime}\Bigl{(}x,v,\rho(x,v)\bigl{(}L(x,y^{\prime}_{\alpha},z_{\alpha})\bigr{)}\Bigr{)}=y^{\prime}_{\alpha}$

as for any $y<y^{\prime}_{\alpha}$ with $\phi(x,y,1)=(x,v)$, the set
$L(x,y,1)$ has strictly less $\rho(x,v)$-measure than
$L(x,y^{\prime}_{\alpha},1)$ as it excludes the atom
$(x,y^{\prime}_{\alpha},z_{\alpha}$). Thus, we get
$(x,y^{\prime}_{\alpha},z_{\alpha})\notin U$.

To show item (c), first note that if $(x,y,z)\in U_{\alpha}$, then we must
have $y>y_{\alpha}$, so there exists $y^{\prime}$ with $y_{\alpha}\leq
y^{\prime}<y$, $\phi(x,y^{\prime},1)=(x,v)$ and
$\rho(x,v)(L(x,y^{\prime},1))\geq M_{\alpha}$ and since $L(x,y,z)\supseteq
L(x,y^{\prime},1)$, we conclude

$\displaystyle\rho(x,v)\bigl{(}L(x,y,z)\bigr{)}$
$\displaystyle\geq\rho(x,v)\bigl{(}L(x,y^{\prime},1)\bigr{)}\geq M_{\alpha}.$

To prove the other inequality, first suppose
$(y,z)<_{L}(y^{\prime}_{\alpha},z_{\alpha})$ and note that the definition of
$z_{\alpha}$ implies that there exists $z^{\prime}\in Z$ such that
$(y,z)<_{L}(y^{\prime}_{\alpha},z^{\prime})$ and
$\rho(x,v)(L(x,y^{\prime}_{\alpha},z^{\prime}))=M_{\alpha}$. Since
$L(x,y,z)\subseteq L(x,y^{\prime}_{\alpha},z^{\prime})$, we get
$\rho(x,v)(L(x,y,z))\leq M_{\alpha}$.

Suppose now that $(y,z)=(y^{\prime}_{\alpha},z_{\alpha})$ and note that the
definition of $z_{\alpha}$ and the facts that $\phi$ does not depend on its
last coordinate and that $(x,y^{\prime}_{\alpha},\widetilde{z}_{\alpha})\in U$
implies $\phi(x,y^{\prime}_{\alpha},\widetilde{z}_{\alpha})=(x,v)$ give us
that for every $z^{\prime}<z_{\alpha}$, we have
$\rho(x,v)(L(x,y^{\prime}_{\alpha},z^{\prime}))\leq M_{\alpha}$. Finally, note
that

$\displaystyle\rho(x,v)\bigl{(}L(x,y,z)\bigr{)}$
$\displaystyle\leq\rho(x,v)(\\{(x,y,z)\\})+\sup_{z^{\prime}<z_{\alpha}}\rho(x,v)\bigl{(}L(x,y^{\prime}_{\alpha},z^{\prime})\bigr{)}\leq
M_{\alpha}$

since $(x,y,z)=(x,y^{\prime}_{\alpha},z_{\alpha})\in U$ and item (b) says that
$(x,y,z)$ is not an atom of $\rho(x,v)$. This concludes the proof of item (c).

Let us now show item (d). The definition of $y_{\alpha}$ and $z_{\alpha}$ as
infimum and supremum, respectively, ensure the existence of a non-increasing
sequence $(y_{\alpha}^{n})_{n\in\mathbb{N}}$ and a non-decreasing sequence
$(z_{\alpha}^{n})_{n\in\mathbb{N}}$ converging to $y_{\alpha}$ and
$z_{\alpha}$, respectively, such that

$\displaystyle\phi(x,y_{\alpha}^{n},1)=\phi(y^{\prime}_{\alpha},z_{\alpha}^{n})=(x,v),$
$\displaystyle\rho(x,v)\bigl{(}L(x,y_{\alpha}^{n},1)\bigr{)}=\rho(x,v)\bigl{(}L(x,y^{\prime}_{\alpha},z_{\alpha}^{n})\bigr{)}=M_{\alpha},$

from which it follows that for every $n\in\mathbb{N}$, letting

$\displaystyle U_{\alpha}^{n}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}L(x,y^{\prime}_{\alpha},z_{\alpha}^{n})\setminus
L(x,y_{\alpha}^{n},1)$ $\displaystyle=\\{(x,y,z)\in X\times Y\times
Z\mid\phi(x,y,z)=(x,v)\land(y_{\alpha}^{n},1)<_{L}(y,z)\leq_{L}(y^{\prime}_{\alpha},z_{\alpha}^{n})\\},$

we get $\rho(x,v)(U_{\alpha}^{n})=0$.

Note now that

$\displaystyle U_{\alpha}$
$\displaystyle\subseteq\bigcup_{n\in\mathbb{N}}U_{\alpha}^{n}\cup\\{(x,y^{\prime}_{\alpha},z_{\alpha})\mid\text{
if }(x,y^{\prime}_{\alpha},z_{\alpha})\in U\\}.$

Since each $U_{\alpha}^{n}$ has zero $\rho(x,v)$-measure and
$(x,y^{\prime}_{\alpha},z_{\alpha})\in U$ implies that it is not an atom of
$\rho(x,v)$ by the already proved item (b), it follows that
$\rho(x,v)(U_{\alpha})=0$, concluding the proof of item (d).

Finally, to show item (e), it suffices to show that if the intervals
$(y_{\beta},y^{\prime}_{\beta})$ and $(y_{\alpha},y^{\prime}_{\alpha})$ have
non-empty intersection, then we have $y^{\prime}_{\alpha}>y^{\prime}_{\beta}$.

Suppose for a contradiction that $y^{\prime}_{\alpha}\leq y^{\prime}_{\beta}$
and that
$(y_{\beta},y^{\prime}_{\beta})\cap(y_{\alpha},y^{\prime}_{\alpha})\neq\varnothing$.
Note that $y^{\prime}_{\alpha}>y_{\beta}$ and since the construction of
$y^{\prime}_{\alpha}$ ensures
$(x,y^{\prime}_{\alpha},\widetilde{z}_{\alpha})\in U\setminus U_{\beta}$, we
must have
$(y^{\prime}_{\alpha},\widetilde{z}_{\alpha})>_{L}(y^{\prime}_{\beta},z_{\beta})$.
Since $y^{\prime}_{\alpha}\leq y^{\prime}_{\beta}$, we get
$y^{\prime}_{\alpha}=y^{\prime}_{\beta}$ and
$\widetilde{z}_{\alpha}>z_{\beta}$.

Since $\widetilde{z}_{\beta}\leq z_{\beta}<\widetilde{z}_{\alpha}\leq
z_{\alpha}$ and $y_{\alpha}<y^{\prime}_{\alpha}=y^{\prime}_{\beta}$ from item
(a), we get $(y^{\prime}_{\beta},\widetilde{z}_{\beta})\in U_{\alpha}$, so
item (c) gives

$\displaystyle M_{\beta}$
$\displaystyle=\rho(x,v)\bigl{(}L(x,y^{\prime}_{\beta},\widetilde{z}_{\beta})\bigr{)}=M_{\alpha},$

which in turn implies $y_{\alpha}=y_{\beta}$. Finally, since
$y^{\prime}_{\alpha}=y^{\prime}_{\beta}$ and $M_{\alpha}=M_{\beta}$, we also
get $z_{\beta}=z_{\alpha}\geq\widetilde{z}_{\alpha}$, contradicting
$\widetilde{z}_{\alpha}>z_{\beta}$.

This concludes the inductive construction.

Note now that item (e) implies that the construction must stop (as the union
of the intervals $(y_{\alpha},y^{\prime}_{\alpha})$ is getting strictly larger
and is always a subset of $[0,1]$), so $\alpha_{0}$ gets defined. We claim
that $\alpha_{0}$ is countable. Indeed, otherwise, for the first uncountable
ordinal $\omega_{1}$, the union
$\bigcup_{\alpha<\omega_{1}}(y_{\alpha},y^{\prime}_{\alpha})\subseteq[0,1]$
has a cover by open sets that does not contain a countable subcover,
contradicting the fact that the usual topology on $[0,1]$ is second-countable.

By construction, we know that $U=\bigcup_{\alpha<\alpha_{0}}U_{\alpha}$, so
item (d) along with the fact that $\alpha_{0}$ is countable implies
$\rho(x,v)(U)=0$. Thus we have proven that $\rho(r)(U)=0$ for every $r\in R$
so item (vi) for $B^{\prime\prime}$
($(\gamma[\mu]\otimes\nu)(B^{\prime\prime})=0$) holds.

Our next objective is to construct $A^{\prime}$ so that the remaining items
(items (i), (vii) and (viii)) hold.

To do so, let us first define for every $(x,v)\in R$ and every $a\in(0,1)$ the
set

$\displaystyle C_{a}(x,v)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\left\\{(x^{\prime},y^{\prime},z^{\prime})\in\phi^{-1}(x,v)\;\middle|\;\rho(x,v)\bigl{(}L(x^{\prime},y^{\prime},z^{\prime})\bigr{)}\leq
a\right\\}.$

We want to show that for $\sigma$-almost every $(x,v)\in R$ we have
$\rho(x,v)(C_{a}(x,v))=a$. For this, we define the following points:

$\displaystyle y_{x,v,a}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\sup\\{y\in[0,1]\mid(x,y,0)\in
C_{a}(x,v)\\}\cup\\{0\\},$ $\displaystyle z_{x,v,a}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\sup\\{z\in[0,1]\mid(x,y_{x,v,a},z)\in
C_{a}(x,v)\\}\cup\\{0\\}$ $\displaystyle
y^{\prime}_{x,v,a}\stackrel{{\scriptstyle\text{def}}}{{=}}G^{\prime}(x,v,a)$
$\displaystyle=\inf\left\\{y\in[0,1]\;\middle|\;\phi(x,v,1)=(x,v)\land\rho(x,v)\bigl{(}L(x,y,1)\bigr{)}\geq
a\right\\}$ $\displaystyle z^{\prime}_{x,v,a}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\inf\left\\{z\in[0,1]\;\middle|\;\phi(x,y^{\prime}_{x,v,a},z)=(x,v)\land\rho(x,v)\bigl{(}L(x,y,1)\bigr{)}\geq
a\right\\}\cup\\{1\\}$

(Note that Claim 8.2.1 along with the fact that $a<1$ ensure that
$y^{\prime}_{x,v,a}$ is actually defined, i.e., that
$(x,v,a)\in\operatorname{dom}(G^{\prime})$.)

We first prove the following weaker claim about the $\rho(x,v)$-measure of
$C_{a}(x,v)$.

###### Claim 8.2.3.

Let $(x,v)\in R$. Then $\rho(x,v)(C_{a}(x,v))\leq a$. Furthermore, if
$\rho(x,v)(C_{a}(x,v))<a$, then $(x,y^{\prime}_{x,v,a},z^{\prime}_{x,v,a})$ is
an atom of $\rho(x,v)$.

###### Proof of Claim 8.2.3..

If $C_{a}(x,v)$ is empty, the we clearly have $\rho(x,v)(C_{a}(x,v))\leq a$.
Suppose then that $C_{a}(x,v)$ is non-empty and note that the definitions of
$y_{x,v,a}$ and $z_{x,v,a}$ as suprema ensure the existence of non-decreasing
sequences $(y_{n})_{n\in\mathbb{N}}$ and $(z_{n})_{n\in\mathbb{N}}$ such that
$(x,y_{n},z_{n})\in C_{a}(x,v)$ and
$C_{a}(x,v)=\bigcup_{n\in\mathbb{N}}L(x,y_{n},z_{n})$ (when any of the suprema
is actually a maximum, the corresponding sequence should be taken constant
equal to the maximum; otherwise, the corresponding sequence should approximate
the supremum).

Since both sequences are non-decreasing, we get $L(x,y_{n},z_{n})\subseteq
L(x,y_{n+1},z_{n+1})$ for every $n\in\mathbb{N}$, so we conclude

$\displaystyle\rho(x,v)\bigl{(}C_{a}(x,v)\bigr{)}$
$\displaystyle=\rho(x,v)\left(\bigcup_{n\in\mathbb{N}}L(x,y_{n},z_{n})\right)=\lim_{n\to\infty}\rho(x,v)(L(x,y_{n},z_{n}))\leq
a,$

where the last inequality follows since $(x,y_{n},z_{n})\in C_{a}(x,v)$.

Suppose now that $\rho(x,v)(C_{a}(x,v))<a$ and note that the definitions of
$y^{\prime}_{x,v,a}$ and $z^{\prime}_{x,v,a}$ as infima ensure the existence
of non-increasing sequences $(y^{\prime}_{n})_{n\in\mathbb{N}}$ and
$(z^{\prime}_{n})_{n\in\mathbb{N}}$ converging to $y^{\prime}_{x,v,a}$ and
$z^{\prime}_{x,v,a}$, respectively and such that
$\phi(x,y^{\prime}_{n},z^{\prime}_{n})=(x,v)$,
$\rho(x,v)(L(x,y^{\prime}_{n},z^{\prime}_{n}))\geq a$ and

$\displaystyle\bigcap_{n\in\mathbb{N}}L(x,y^{\prime}_{n},z^{\prime}_{n})$
$\displaystyle=\left\\{(x,y,z)\;\middle|\;\phi(x,y,z)=(x,v)\land\rho(x,v)\bigl{(}L(x,y,z)\bigr{)}\geq
a\right\\}.$

(Again, whenever any of the infima is actually a minimum, the corresponding
sequence should be taken constant equal to the minimum.)

Since both sequences are non-increasing, we get
$L(x,y^{\prime}_{n},z^{\prime}_{n})\supseteq
L(x,y^{\prime}_{n+1},z^{\prime}_{n+1})$, from which we conclude

$\displaystyle\rho(x,v)\left(\bigcap_{n\in\mathbb{N}}L(x,y^{\prime}_{n},z^{\prime}_{n})\right)$
$\displaystyle=\lim_{n\to\infty}\rho(x,v)\bigl{(}L(x,y^{\prime}_{n},z^{\prime}_{n})\bigr{)}\geq
a.$

On the other hand, since $y^{\prime}_{n}$ and $z^{\prime}_{n}$ converge to
$y^{\prime}_{x,v,a}$ and $z^{\prime}_{x,v,a}$ respectively, we have

$\displaystyle\bigcap_{n\in\mathbb{N}}L(x,y^{\prime}_{n},z^{\prime}_{n})$
$\displaystyle=\\{(x^{\prime},y^{\prime},z^{\prime})\in\phi^{-1}(x,v)\mid(y^{\prime},z^{\prime})\leq_{L}(y^{\prime}_{x,v,a},z^{\prime}_{x,v,a})\\}.$

Let $C^{\prime}_{a}$ be the set above and note that since
$\rho(x,v)(C_{a}(x,v))<a\leq\rho(x,v)(C^{\prime}_{a})$, we must have
$C_{a}(x,v)\subsetneq C^{\prime}_{a}$ (since both these sets are subsets of
$\phi^{-1}(x,v)$ that are closed downward with respect to $\leq_{L}$ within
$\phi^{-1}(x,v)$).

We claim that $C^{\prime}_{a}\setminus
C_{a}(x,v)=\\{(x,y^{\prime}_{x,v,a},z^{\prime}_{x,v,a})\\}$. To see this, let
$(x^{\prime},y^{\prime},z^{\prime})\in C^{\prime}_{a}\setminus C_{a}(x,v)$ and
note that the fact that $(x^{\prime},y^{\prime},z^{\prime})\notin C_{a}(x,v)$
gives $\rho(x,v)(L(x^{\prime},y^{\prime},z^{\prime}))>a$. From
$\phi(x^{\prime},y^{\prime},z^{\prime})=(x,v)$ along with the definitions of
$y^{\prime}_{x,v,a}$ and $z^{\prime}_{x,v,a}$ as infima, we get $x^{\prime}=x$
and $(y^{\prime},z^{\prime})\geq_{L}(y^{\prime}_{x,v,a},z^{\prime}_{x,v,a})$.
In turn, since $(x^{\prime},y^{\prime},z^{\prime})\in C^{\prime}_{a}$, we
conclude
$(y^{\prime},z^{\prime})\leq_{L}(y^{\prime}_{x,v,a},z^{\prime}_{x,v,a})$.
Thus,
$(x^{\prime},y^{\prime},z^{\prime})=(x,y^{\prime}_{x,v,a},z^{\prime}_{x,v,a})$,
hence $C^{\prime}_{a}\setminus
C_{a}(x,v)=\\{(x,y^{\prime}_{x,v,a},z^{\prime}_{x,v,a})\\}$.

Finally, since

$\displaystyle\rho(x,v)(\\{x,y^{\prime}_{x,v,a},z^{\prime}_{x,v,a}\\})$
$\displaystyle=\rho(x,v)(C^{\prime}_{a}\setminus C_{a}(x,v))>0,$

we conclude that $(x,y^{\prime}_{x,v,a},z^{\prime}_{x,v,a})$ is an atom of
$\rho(x,v)$. ∎

For every $a\in(0,1)$, let

$\displaystyle S_{a}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\\{(x,y,z)\in
B^{\prime\prime}\mid y=y^{\prime}_{\phi(x,y,z),a}\land
z=z^{\prime}_{\phi(x,y,z),a}\\}.$

Since $\nu=\lambda$ is atomless and for each $(x,y)$ there exists at most one
$z$ such that $(x,y,z)\in S_{a}$, by Fubini’s Theorem, the set $S_{a}$ must
have zero $\gamma[\mu]\otimes\nu$-measure. On the other hand, by (8.1) with
$h\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbbm{1}_{S_{a}}$, we have

$\displaystyle 0$
$\displaystyle=(\gamma[\mu]\otimes\nu)(S_{a})=\int_{R}\int_{\phi^{-1}(x,v)}\mathbbm{1}_{S_{a}}(x^{\prime},y^{\prime},z^{\prime})\
d\rho(x,v)(x^{\prime},y^{\prime},z^{\prime})\ d\sigma(x,v)$
$\displaystyle=\int_{R}\rho(x,v)(\\{(x,y^{\prime}_{x,v,a},z^{\prime}_{x,v,a})\\})\
d\sigma(x,v),$

from which we conclude that for $\sigma$-almost every $(x,v)\in R$, we have
$\rho(x,v)(\\{x,y^{\prime}_{x,v,a},z^{\prime}_{x,v,a}\\})=0$, which along with
Claim 8.2.3 allows us to conclude that

$\displaystyle\rho(x,v)(C_{a}(x,v))=a$

for $\sigma$-almost every $(x,v)\in R$.

We now define

$\displaystyle R^{\prime}_{a}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\left\\{(x,v)\in
R\;\middle|\;\rho(x,v)\bigl{(}C_{a}(x,v)\bigr{)}=a\right\\}\qquad\bigl{(}a\in(0,1)\bigr{)}$
$\displaystyle R^{\prime}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\bigcap_{a\in(0,1)\cap\mathbb{Q}}R^{\prime}_{a}.$

We have just proved that $\sigma(R^{\prime}_{a})=1$ for all $a\in(0,1)$, which
also implies $\sigma(R^{\prime})=1$.

In particular, we get

$\displaystyle 1=\sigma(R^{\prime})$
$\displaystyle=\int_{R}\mathbbm{1}_{R^{\prime}}(r)\ d\sigma(r)$
$\displaystyle=\int_{R}\int_{\phi^{-1}(r)}\mathbbm{1}_{\phi^{-1}(R^{\prime})}(s)\
d\rho(r)(s)\ d\sigma(r)$ $\displaystyle=\int_{X\times Y\times
Z}\mathbbm{1}_{\phi^{-1}(R^{\prime})}(s)\ d(\gamma[\mu]\otimes\nu)(s)$
$\displaystyle=\int_{X}(\gamma(x)\otimes\nu)(\\{(y,z)\in Y\times
Z\mid\phi(x,y,z)\in R^{\prime}\\})\ d\mu(x)$

where the second equality follows since $\sigma(R)=1$, the third equality
follows since when $r\in R$, we have $\rho(r)(\phi^{-1}(r))=1$, the third
equality follows from (8.1) with
$h\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbbm{1}_{\phi^{-1}(R^{\prime})}$
and the last equality follows from Fubini’s Theorem.

Thus, we conclude that defining

$\displaystyle A^{\prime}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\left\\{x\in
A\;\middle|\;(\gamma(x)\otimes\nu)\bigl{(}\\{(y,z)\in Y\times
Z\mid\phi(x,y,z)\in R^{\prime}\\}\bigr{)}=1\right\\},$

we have $\mu(A^{\prime})=1$, that is, item (vii) is satisfied. It is also
obvious from the definition of $A^{\prime}$ that item (i)
($A^{\prime}\subseteq A$) holds.

It remains to show item (viii), that is, we want to show that for every $x\in
A^{\prime}$, we have
$G(x,\mathord{-},\mathord{-})_{*}(\gamma(x)\otimes\nu)=\Theta(f(x))$. To do
so, we pick $(\bm{y},\bm{z})$ according to $\gamma(x)\otimes\nu$, we let
$(\bm{v},\bm{w})\stackrel{{\scriptstyle\text{def}}}{{=}}G(x,\bm{y},\bm{z})$
and we need to show that $(\bm{v},\bm{w})$ is distributed according to
$\Theta(f(x))=\theta(f(x))\otimes\eta$ (recall $\eta=\lambda$).

The hypothesis $g(x,\mathord{-})_{*}(\gamma(x))=\theta(f(x))$ for every $x\in
A$ along with the already proved items (i) and (v) ($A^{\prime}\subseteq A$
and $\pi(G(x,y,z))=g(x,y)$) implies that $\bm{v}$ is distributed according to
$\theta(f(x))$. On the other hand, the definition of $G$ in (8.2) implies that
for every $a\in[0,1]$, we have the following conditional probability equality:

$\displaystyle\mathbb{P}[\bm{w}\leq a\mathbin{|}\bm{v}]$
$\displaystyle=\mathbb{P}\bigl{[}\rho(x,\bm{v})(L(x,\bm{y},\bm{z}))\leq
a\mathbin{\big{|}}\bm{v}\bigr{]}=\rho(x,\bm{v})\bigl{(}C_{a}(x,\bm{v})\bigr{)}.$

On the other hand, by the definition of $A^{\prime}$, we know that for every
$a\in(0,1)\cap\mathbb{Q}$, we have

$\displaystyle\mathbb{P}\Bigl{[}\rho(x,\bm{v})\bigl{(}C_{a}(x,\bm{v})\bigr{)}=a\Bigr{]},$

from which we get that the conditional distribution of $\bm{w}$ given $\bm{v}$
is $\lambda$ (with probability $1$ over $\bm{v}$), so $(\bm{w},\bm{v})$ has
distribution $\theta(f(x))\otimes\lambda=\Theta(f(x))$, as desired.

Finally, as mentioned after the definition of $B^{\prime\prime}$ in (8.4),
letting
$B^{\prime}\stackrel{{\scriptstyle\text{def}}}{{=}}B^{\prime\prime}\cap(A^{\prime}\times
Y\times Z)$ concludes the proof in the case when
$(X,\mu)=(Z,\nu)=([0,1],\lambda)$ and $W=Y=[0,1]$.

We now prove that the general case reduces to the previous case. Let
$I_{X}\colon[0,1]\to X$, $I_{Z}\colon[0,1]\to Z$ and $I_{W}\colon[0,1]\to W$
be measure-isomorphisms modulo $0$ from $([0,1],\lambda)$ to $(X,\mu)$,
$(Z,\nu)$ and $(W,\eta)$, respectively. Since $Y$ is uncountable, there exists
a Borel-isomorphism $J_{Y}\colon[0,1]\to Y$.

Let now $\widetilde{f}\colon[0,1]\rightharpoonup U$,
$\widetilde{g}\colon[0,1]\times[0,1]\rightharpoonup V$,
$\widetilde{\gamma}\colon[0,1]\rightharpoonup{\mathcal{P}}([0,1])$,
$\widetilde{A}\subseteq[0,1]$ and $\widetilde{B}\subseteq[0,1]\times[0,1]$ be
given by

$\displaystyle\begin{aligned}
\widetilde{f}&\stackrel{{\scriptstyle\text{def}}}{{=}}f\mathbin{\circ}I_{X},&\widetilde{g}&\stackrel{{\scriptstyle\text{def}}}{{=}}g\mathbin{\circ}(I_{X}\otimes
J_{Y}),\\\
\widetilde{A}&\stackrel{{\scriptstyle\text{def}}}{{=}}I_{X}^{-1}(A),&\widetilde{B}&\stackrel{{\scriptstyle\text{def}}}{{=}}(I_{X}\otimes
J_{Y})^{-1}(B),\end{aligned}$
$\displaystyle\widetilde{\gamma}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}(J_{Y}^{-1})_{*}\Bigl{(}\gamma\bigl{(}I_{X}(x)\bigr{)}\Bigr{)}.$

Let then $\widetilde{G}\colon[0,1]\times[0,1]\times[0,1]\rightharpoonup
V\times[0,1]$, $\widetilde{G}^{\prime}\colon[0,1]\times
V\times[0,1]\rightharpoonup[0,1]$,
$\widetilde{A}^{\prime}\subseteq\widetilde{A}$ and
$\widetilde{B}^{\prime}\subseteq\widetilde{B}\times[0,1]$ be the
$(G,G^{\prime},A^{\prime},B^{\prime})$ provided by the lemma for
$(\widetilde{f},\widetilde{g},\widetilde{\gamma},\widetilde{A},\widetilde{B})$
(it is straightforward to check that the hypotheses are satisfied and this is
the already proved case).

Then define

$\displaystyle G$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(\operatorname{id}_{V}\otimes
I_{W})\mathbin{\circ}\widetilde{G}\mathbin{\circ}(I_{X}^{-1}\otimes
J_{Y}^{-1}\otimes I_{Z}^{-1}),$ $\displaystyle G^{\prime}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}J_{Y}\mathbin{\circ}\widetilde{G}^{\prime}\mathbin{\circ}(I_{X}^{-1}\otimes\operatorname{id}_{V}\otimes
I_{W}^{-1}),$ $\displaystyle A^{\prime}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}I_{X}(\widetilde{A}^{\prime})\cap\operatorname{dom}(I_{X}^{-1}),$
$\displaystyle B^{\prime}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(I_{X}\otimes
J_{Y}\otimes
I_{Z})(\widetilde{B}^{\prime})\cap(\operatorname{dom}(I_{X}^{-1})\times
Y\times\operatorname{dom}(I_{Z}^{-1})),$

It is straightforward to check that the conclusions of the lemma for
$(G,G^{\prime},A^{\prime},B^{\prime})$ follow from the conclusions of the
lemma for
$(\widetilde{G},\widetilde{G}^{\prime},\widetilde{A}^{\prime},\widetilde{B}^{\prime})$.
∎

The next lemma is the key lemma of this section that will be used to formalize
the construction of the amalgamating types. It upgrades Lemma 8.2 in two ways:
the first is so that the resulting $G^{\prime}(x,\mathord{-},\mathord{-})$ is
an almost everywhere left-inverse of $G(x,\mathord{-},\mathord{-})$ (i.e., it
is able to retrieve both the $Y$ and $Z$ coordinates instead of only the
former); the second is an added equivariant version of the construction.

###### Lemma 8.3.

Let $(X,\mu)$, $(Z,\nu)$ and $(W,\eta)$ be atomless standard probability
spaces, let $Y$ and $U$ be Borel spaces with $Y$ uncountable, let $f\colon
X\rightharpoonup U$ and $g\colon X\times Y\rightharpoonup V$ be measurable
partial functions, let $\gamma\colon X\rightharpoonup{\mathcal{P}}(Y)$ and
$\theta\colon U\rightharpoonup{\mathcal{P}}(V)$ be partial Markov kernels with
$\mu(\operatorname{dom}(\gamma))=1$ and let
$A\subseteq\operatorname{dom}(f)\subseteq X$ and
$B\subseteq\operatorname{dom}(g)\subseteq X\times Y$ be measurable sets with
$\mu(A)=\gamma[\mu](B)=1$ (see (6.1)). Suppose further that $A\subseteq
f^{-1}(\operatorname{dom}(\theta))\cap\operatorname{dom}(\gamma)$ and for
every $x\in A$, we have

$\displaystyle g(x,\mathord{-})_{*}(\gamma(x))$ $\displaystyle=\theta(f(x)).$

Finally, let $\pi\colon V\times W\to V$ be the natural projection.

Then there exist partial functions $G\colon X\times Y\times Z\rightharpoonup
V\times W$ and $G^{\prime}\colon X\times V\times W\rightharpoonup Y\times Z$
and measurable sets $A^{\prime}\subseteq X$ and $B^{\prime}\subseteq X\times
Y\times Z$ such that letting $\Theta\colon
U\rightharpoonup{\mathcal{P}}(V\times W)$ be the partial Markov kernel given
by

$\displaystyle\Theta(u)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\theta(u)\otimes\eta\qquad(u\in
U),$

the following hold:

  1. i.

$A^{\prime}\subseteq A$.

  2. ii.

$B^{\prime}\subseteq((A^{\prime}\times Y)\cap B)\times Z$.

  3. iii.

$B^{\prime}\subseteq\operatorname{dom}(G)$.

  4. iv.

For every $(x,y,z)\in B^{\prime}$, we have $G^{\prime}(x,G(x,y,z))=(y,z)$.

  5. v.

For every $(x,y,z)\in B^{\prime}$, we have $\pi(G(x,y,z))=g(x,y)$.

  6. vi.

$(\gamma[\mu]\otimes\nu)(B^{\prime})=1$.

  7. vii.

$\mu(A^{\prime})=1$.

  8. viii.

For every $x\in A^{\prime}$, we have
$G(x,\mathord{-},\mathord{-})_{*}(\gamma(x)\otimes\nu)=\Theta(f(x))$.

Furthermore, if $H$ is a finite group such that the following hold:

  * •

$H$ acts measurably on $X$, $Y$, $Z$, $U$, $V$ and $W$.

  * •

There exists a measurable set $X^{\prime}\subseteq X$ such that $H$ the orbits
$(\sigma\cdot X^{\prime})_{\sigma\in H}$ of $X^{\prime}$ are pairwise disjoint
and $\mu(\bigcup_{\sigma\in H}\sigma\cdot X^{\prime})=1$.

  * •

The $H$-actions on $Z$ and $W$ are trivial.

  * •

$\mu$ is $H$-invariant.

  * •

$f$, $g$, $\gamma$ and $\theta$ are $H$-equivariant (when we equip product
spaces with the diagonal action and ${\mathcal{P}}(V)$ and ${\mathcal{P}}(Y)$
with the induced actions).

Then $G$, $G^{\prime}$, $\Theta$, $A^{\prime}$ and $B^{\prime}$ can be taken
to further satisfy:

  1. ix.

$A^{\prime}$ and $B^{\prime}$ are $H$-invariant (with respect to the diagonal
actions).

  2. x.

$G$, $G^{\prime}$ and $\Theta$ are $H$-equivariant (with respect to the
diagonal actions and the induced action on ${\mathcal{P}}(V\times W)$).

###### Proof.

We will first show all the statements that do not involve the group $H$.

We first prove the case when
$(Z,\nu)=(\widetilde{Z}^{\mathbb{N}_{+}},\widetilde{\nu}^{\mathbb{N}_{+}})$
and
$(W,\eta)=(\widetilde{W}^{\mathbb{N}_{+}},\widetilde{\eta}^{\mathbb{N}_{+}})$
for some atomless standard probability spaces
$(\widetilde{Z},\widetilde{\nu})$ and $(\widetilde{W},\widetilde{\eta})$.

For every $m,n\in\mathbb{N}$ with $m\geq n$, let
$\tau_{m,n}\colon\widetilde{Z}^{m}\to\widetilde{Z}^{n}$,
$\tau_{\mathbb{N}_{+},n}\colon\widetilde{Z}^{\mathbb{N}_{+}}\to\widetilde{Z}^{n}$,
$\pi_{m,n}\colon\widetilde{W}^{m}\to\widetilde{W}^{n}$ and
$\pi_{\mathbb{N}_{+},n}\colon\widetilde{W}^{\mathbb{N}_{+}}\to\widetilde{W}^{n}$
be the projections onto the first $n$ coordinates.

Define

$\displaystyle G_{-1}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}g,$
$\displaystyle\Gamma_{-1}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\gamma,$
$\displaystyle\Theta_{-1}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\theta,$ $\displaystyle
A_{-1}$ $\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}A,$
$\displaystyle B_{-1}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}B$

and let us construct inductively in $n\in\mathbb{N}$ sequences of measurable
partial functions $G_{n}\colon X\times
Y\times\widetilde{Z}^{n+1}\rightharpoonup V\times\widetilde{W}^{n+1}$ and
$G^{\prime}_{n}\colon X\times V\times\widetilde{W}^{n+1}\rightharpoonup
X\times Y\times\widetilde{Z}^{n}$, sequences of partial Markov kernels
$\Gamma_{n}\colon X\rightharpoonup{\mathcal{P}}(Y\times\widetilde{Z}^{n+1})$
and $\Theta_{n}\colon
U\rightharpoonup{\mathcal{P}}(V\times\widetilde{W}^{n+1})$ and sequences of
measurable sets $A_{n}\subseteq X$ and $B_{n}\subseteq X\times
Y\times\widetilde{Z}^{n+1}$ such that the following hold:

  1. I.

$A_{n}\subseteq A_{n-1}$.

  2. II.

$B_{n}\subseteq((A_{n}\times Y\times\widetilde{Z}^{n})\cap
B_{n-1})\times\widetilde{Z}$.

  3. III.

$B_{n}\subseteq\operatorname{dom}(G_{n})$.

  4. IV.

For every $(x,y,z)\in B_{n}$, we have
$G^{\prime}_{n}(x,G_{n}(x,y,z))=(y,\tau_{n+1,n}(z))$.

  5. V.

For every $(x,y,z)\in B_{n}$, we have
$(\operatorname{id}_{V}\otimes\pi_{n+1,n})(G_{n}(x,y,z))=G_{n-1}(x,y,\tau_{n+1,n}(z))$.

  6. VI.

$\Gamma_{n}[\mu](B_{n})=1$.

  7. VII.

$\mu(A_{n})=1$.

  8. VIII.

For every $x\in A_{n}$, we have
$G_{n}(x,\mathord{-},\mathord{-})_{*}(\Gamma_{n}(x))=\Theta_{n}(f(x))$.

  9. IX.

$A_{n}\subseteq\operatorname{dom}(f)$.

  10. X.

$\mu(\operatorname{dom}(\Gamma_{n}))=1$.

Assume that we are given $(G_{n-1},\Gamma_{n-1},\Theta_{n-1},A_{n-1},B_{n-1})$
satisfying items (III), (VI), (VII), (VIII), (IX) and (X) (note that these
indeed hold for $(G_{-1},\Gamma_{-1},\Theta_{-1},A_{-1},B_{-1})$) and note
that the hypotheses of Lemma 8.2 are satisfied with the following choice of
parameters:

$\displaystyle(X,\mu)$ $\displaystyle\leftarrow(X,\mu),$
$\displaystyle(Z,\nu)$
$\displaystyle\leftarrow(\widetilde{Z},\widetilde{\nu}),$ $\displaystyle Y$
$\displaystyle\leftarrow Y\times\widetilde{Z}^{n},$ $\displaystyle U$
$\displaystyle\leftarrow U,$ $\displaystyle V$ $\displaystyle\leftarrow
V\times\widetilde{W}^{n},$ $\displaystyle(W,\eta)$
$\displaystyle\leftarrow(\widetilde{W},\widetilde{\eta}),$ $\displaystyle f$
$\displaystyle\leftarrow f,$ $\displaystyle g$ $\displaystyle\leftarrow
G_{n-1},$ $\displaystyle\gamma$ $\displaystyle\leftarrow\Gamma_{n-1},$
$\displaystyle\theta$ $\displaystyle\leftarrow\Theta_{n-1},$ $\displaystyle A$
$\displaystyle\leftarrow A_{n-1},$ $\displaystyle B$ $\displaystyle\leftarrow
B_{n-1}.$

Let then $(G_{n},G^{\prime}_{n},\Theta_{n},A_{n},B_{n})$ be the
$(G,G^{\prime},\Theta,A^{\prime},B^{\prime})$ provided by Lemma 8.2 with the
choice of parameters above. Define also the partial Markov kernel
$\Gamma_{n}\colon X\rightharpoonup{\mathcal{P}}(Y\times\widetilde{Z}^{n+1})$
by

$\displaystyle\Gamma_{n}(x)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\Gamma_{n-1}(x)\otimes\widetilde{\nu}\qquad(x\in
X)$

(which by induction is equivalent to
$\Gamma_{n}(x)=\gamma(x)\otimes\widetilde{\nu}^{n+1}$). Note that we have the
following implications between the items of Lemma 8.2 and the items required
by our inductive construction:

(i)$\implies$ |  (I), |  (ii)$\implies$ |  (II), |  (iii)$\implies$ |  (III), |  (iv)$\implies$ |  (IV),
---|---|---|---|---|---|---|---
(v)$\implies$ |  (V), |  (vi)$\implies$ |  (VI), |  (vii)$\implies$ |  (VII), |  (viii)$\implies$ |  (VIII).

Note also that item (IX) follows from the same item for $A_{n-1}$ and the
already proven item (I) and item (X) follows from the same item for
$\Gamma_{n-1}$.

Define then the sets

$\displaystyle A^{\prime}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\bigcap_{n\in\mathbb{N}}A_{n},$
$\displaystyle B^{\prime}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\bigcap_{n\in\mathbb{N}}(\operatorname{id}_{X}\otimes\operatorname{id}_{Y}\otimes\tau_{\mathbb{N}_{+},n})^{-1}(B_{n-1}).$

Since item (I) implies $A_{n}\subseteq A_{-1}=A$ for every $n\in\mathbb{N}$,
item (i) ($A^{\prime}\subseteq A)$ holds.

An inductive application of item (II) shows that item (ii)
($B^{\prime}\subseteq((A^{\prime}\times Y)\cap B)\times Z$) holds.

Define the measurable partial function $G\colon X\times Y\times
Z\rightharpoonup V\times W$ by letting

$\displaystyle G(x,y,z)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\bigl{(}g(x,y),w\bigr{)}\qquad(x\in
X,y\in Y,z\in Z),$ $\displaystyle w_{n}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}G_{n-1}(x,y,\tau_{\mathbb{N}_{+},n}(z))_{n}\qquad(n\in\mathbb{N}_{+}).$

Note that item (III) implies item (iii)
($B^{\prime}\subseteq\operatorname{dom}(G)$) holds.

Define now the measurable partial function $G^{\prime}\colon X\times V\times
W\rightharpoonup Y\times Z$ by letting

$\displaystyle G^{\prime}(x,v,w)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\Bigl{(}G^{\prime}_{0}\bigl{(}x,v,\pi_{\mathbb{N}_{+},1}(w)\bigr{)},z)\qquad(x\in
X,v\in V,w\in W),$ $\displaystyle z_{n}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\widetilde{\tau}_{n}\Bigl{(}G^{\prime}_{n}\bigl{(}x,v,\pi_{\mathbb{N}_{+},n+1}(w)\bigr{)}\Bigr{)}\qquad(n\in\mathbb{N}),$

where $\widetilde{\tau}_{n}\colon Y\times\widetilde{Z}^{n}\to\widetilde{Z}$ is
the projection onto the last coordinate of $\widetilde{Z}^{n}$. A simple
induction shows that item (IV) implies item (iv)
($G^{\prime}(x,(G(x,y,z)))=(y,z)$ whenever $(x,y,z)\in B^{\prime}$) holds.

It is obvious from the definition of $G$ that item (v) ($\pi(G(x,y,z))=g(x,y)$
whenever $(x,y,z)\in B^{\prime}$) holds.

Since $\Gamma_{n}(x)=\gamma(x)\otimes\widetilde{\nu}^{n+1}$, an inductive
application of item (VI) implies item (vi)
($(\gamma[\mu]\otimes\nu)(B^{\prime})=1$) holds.

It is also clear that item (VII) implies that item (vii) ($\mu(A^{\prime})=1$)
holds.

Recalling now that the partial Markov kernel $\Theta\colon
U\rightharpoonup{\mathcal{P}}(V\times W)$ is given by

$\displaystyle\Theta(u)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\theta(u)\otimes\eta=\theta(u)\otimes\widetilde{\eta}^{\mathbb{N}_{+}}\qquad(u\in
U),$

by item (VIII), it follows that item (viii)
($G(x,\mathord{-},\mathord{-})_{*}(\gamma(x)\otimes\nu)=\Theta(f(x))$) holds.

Let us now prove that the general case reduces to the previous case. Let
$I_{Z}\colon Z^{\mathbb{N}_{+}}\to Z$ and $I_{W}\colon W^{\mathbb{N}_{+}}\to
W$ be measure-isomorphisms modulo $0$ from
$(Z^{\mathbb{N}_{+}},\nu^{\mathbb{N}_{+}})$ to $(Z,\nu)$ and from
$(W^{\mathbb{N}_{+}},\eta^{\mathbb{N}_{+}})$ to $(W,\eta)$.

Let $\widetilde{G}\colon X\times Y\times Z^{\mathbb{N}_{+}}\rightharpoonup
V\times W^{\mathbb{N}_{+}}$, $\widetilde{G}^{\prime}\colon X\times V\times
W^{\mathbb{N}+}\rightharpoonup Y\times Z^{\mathbb{N}_{+}}$,
$\widetilde{A}^{\prime}\subseteq A$, $\widetilde{B}^{\prime}\subseteq B\times
Z^{\mathbb{N}_{+}}$ be the $(G,G^{\prime},A^{\prime},B^{\prime})$ provided by
the lemma in the previous case.

It is then straightforward to check that setting

$\displaystyle\begin{aligned}
G&\stackrel{{\scriptstyle\text{def}}}{{=}}(\operatorname{id}_{V}\otimes
I_{W})\mathbin{\circ}\widetilde{G}\mathbin{\circ}(\operatorname{id}_{X}\otimes\operatorname{id}_{Y}\otimes
I_{Z}^{-1}),&G^{\prime}&\stackrel{{\scriptstyle\text{def}}}{{=}}(\operatorname{id}_{Y}\otimes
I_{Z})\mathbin{\circ}\widetilde{G}^{\prime}\mathbin{\circ}(\operatorname{id}_{X}\otimes\operatorname{id}_{V}\otimes
I_{W}^{-1}),\end{aligned}$ $\displaystyle\begin{aligned}
A^{\prime}&\stackrel{{\scriptstyle\text{def}}}{{=}}\widetilde{A}^{\prime},&B^{\prime}&\stackrel{{\scriptstyle\text{def}}}{{=}}(\operatorname{id}_{X}\otimes\operatorname{id}_{Y}\otimes
I_{Z})(\widetilde{B}^{\prime})\cap(X\times
Y\times\operatorname{dom}(I_{Z}^{-1}))\end{aligned}$

gives the desired result.

We now prove that the equivariant version of the lemma involving the group $H$
reduces to the non-equivariant version.

First let

$\displaystyle A_{H}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\bigcap_{\sigma\in
H}(\sigma\cdot A)\cap X^{\prime},$ $\displaystyle B_{H}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\bigcap_{\sigma\in
H}(\sigma\cdot B)\cap(X^{\prime}\times Y).$

It will also be convenient to set
$X_{H}\stackrel{{\scriptstyle\text{def}}}{{=}}\bigcup_{\sigma\in H}\sigma\cdot
X^{\prime}$.

Note that since the orbits $(\sigma\cdot X^{\prime})_{\sigma\in H}$ of
$X^{\prime}$ are pairwise disjoint and $\mu(X_{H})=1$, it follows that

$\displaystyle\mu(A_{H})=\gamma[\mu](B_{H})=\mu(X^{\prime})$
$\displaystyle=\frac{1}{\lvert H\rvert}.$

Note that for all measurable $C\subseteq X$ and all measurable $D\subseteq Y$,
we have

$\displaystyle\gamma[\mu](C\times D)$ $\displaystyle=\int_{C}\gamma(x)(D)\
d\mu(x)=\int_{C}\gamma(\sigma\cdot x)(\sigma\cdot D)\ d\mu(x)$
$\displaystyle=\int_{\sigma\cdot C}\gamma(z)(\sigma\cdot D)\
d\mu(z)=\gamma[\mu](\sigma\cdot(C\times D)),$

where the second equality follows since $\gamma$ is $H$-equivariant and the
third equality follows since $\mu$ is $H$-invariant. Thus $\gamma[\mu]$ is
$H$-invariant. In particular, since $\gamma[\mu](B)=1$, it follows that
$\gamma[\mu](\bigcap_{\sigma\in H}\sigma\cdot B)=1$.

We now let $\mu_{H}$ be the probability measure on $X^{\prime}$ that is the
rescaled restriction of $\mu$, that is,
$\mu_{H}\stackrel{{\scriptstyle\text{def}}}{{=}}\lvert
H\rvert\cdot\mu\mathord{|}_{X^{\prime}}$. It is also clear from definitions
that $\gamma\mathord{|}_{X^{\prime}}[\mu_{H}]=\lvert
H\rvert\cdot\gamma[\mu]\mathord{|}_{X^{\prime}\times Y}$. In particular, we
have

$\displaystyle\gamma\mathord{|}_{X^{\prime}}[\mu_{H}](B_{H})$
$\displaystyle=\lvert H\rvert\cdot\gamma[\mu]\left(\bigcap_{\sigma\in
H}(\sigma\cdot B)\cap(X^{\prime}\times Y)\right)=\lvert
H\rvert\cdot\gamma[\mu](X^{\prime}\times Y)=1,$

where the second equality follows since $\gamma[\mu](\bigcap_{\sigma\in
H}\sigma\cdot B)=1$ as $\gamma[\mu](B)=1$ and $\gamma[\mu]$ is $H$-invariant.

This means that we can apply the current lemma without the
equivariance/invariance conclusions with the following choice of parameters:

$\displaystyle(X,\mu)$ $\displaystyle\leftarrow(X^{\prime},\mu_{H}),$
$\displaystyle(Z,\nu)$ $\displaystyle\leftarrow(Z,\nu),$ $\displaystyle Y$
$\displaystyle\leftarrow Y,$ $\displaystyle U$ $\displaystyle\leftarrow U,$
$\displaystyle V$ $\displaystyle\leftarrow V,$ $\displaystyle(W,\eta)$
$\displaystyle\leftarrow(W,\eta),$ $\displaystyle f$ $\displaystyle\leftarrow
f\mathord{|}_{X^{\prime}},$ $\displaystyle g$ $\displaystyle\leftarrow
g\mathord{|}_{X^{\prime}\times Y},$ $\displaystyle\gamma$
$\displaystyle\leftarrow\gamma\mathord{|}_{X^{\prime}}$ $\displaystyle\theta$
$\displaystyle\leftarrow\theta,$ $\displaystyle A$ $\displaystyle\leftarrow
A_{H},$ $\displaystyle B$ $\displaystyle\leftarrow B_{H}.$

Let then $(G_{H},G^{\prime}_{H},A^{\prime}_{H},B^{\prime}_{H})$ be the
$(G,G^{\prime},A^{\prime},B^{\prime})$ provided by the lemma with the choice
of parameters above.

We then define $(G,G^{\prime},A^{\prime},B^{\prime})$ by

$\displaystyle G(x,y,z)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\tau_{x}^{-1}\cdot
G_{H}\bigl{(}\tau_{x}\cdot(x,y,z)\bigr{)}\qquad(x\in X_{H},y\in Y,z\in Z)$
$\displaystyle G^{\prime}(x,v,w)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\tau_{x}^{-1}\cdot
G^{\prime}_{H}\bigl{(}\tau_{x}\cdot(x,v,w)\bigr{)},\qquad(x\in X_{H},v\in
V,w\in W),$ $\displaystyle A^{\prime}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\bigcup_{\sigma\in
H}\sigma\cdot A^{\prime}_{H},$ $\displaystyle B^{\prime}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\bigcup_{\sigma\in
H}\sigma\cdot B^{\prime}_{H},$

where $\tau_{x}\in H$ is the unique group element such that $\tau_{x}\cdot
x\in X^{\prime}$ (recall that the orbits $(\sigma\cdot X^{\prime})_{\sigma\in
H}$ of $X^{\prime}$ are pairwise disjoint).

It is obvious that $A^{\prime}$ and $B^{\prime}$ are $H$-invariant (i.e., item
(ix) holds). Note also that for $x\in X_{H}$ and $\sigma\in H$, we have
$\tau_{\sigma\cdot x}=\tau_{x}\mathbin{\circ}\sigma^{-1}$, from which it
follows that

$\displaystyle G\bigl{(}\sigma\cdot(x,y,z)\bigr{)}$
$\displaystyle=\tau_{\sigma\cdot x}^{-1}\cdot G_{H}\bigl{(}(\tau_{\sigma\cdot
x}\mathbin{\circ}\sigma)\cdot(x,y,z)\bigr{)}=\sigma\cdot\Bigl{(}\tau_{x}^{-1}\cdot
G_{H}\bigl{(}\tau_{x}\cdot(x,y,z)\bigr{)}\Bigr{)}=\sigma\cdot G(x,y,z),$
$\displaystyle G^{\prime}\bigl{(}\sigma\cdot(x,v,w)\bigr{)}$
$\displaystyle=\tau_{\sigma\cdot x}^{-1}\cdot
G^{\prime}_{H}\bigl{(}(\tau_{\sigma\cdot
x}\mathbin{\circ}\sigma)\cdot(x,v,w)\bigr{)}=\sigma\cdot\Bigl{(}\tau_{x}^{-1}\cdot
G^{\prime}_{H}\bigl{(}\tau_{x}\cdot(x,v,w)\bigr{)}\Bigr{)}=\sigma\cdot
G^{\prime}(x,v,w),$

for every $y\in Y$, every $z\in Z$, every $v\in V$ and every $w\in W$. Thus,
$G$ and $G^{\prime}$ are $H$-equivariant. Finally, the definition of $\Theta$
along with the fact that $\theta$ is $H$-equivariant and $\eta$ is
$H$-invariant (as the $H$-action on $W$ is trivial) implies that $\Theta$ is
$H$-equivariant. Thus, item (x) holds.

Let us now check that all items not concerning invariance/equivariance follow
from their counterparts for
$(G_{H},G^{\prime}_{H},A^{\prime}_{H},B^{\prime}_{H})$.

Item (i) is straightforward:

$\displaystyle A^{\prime}$ $\displaystyle=\bigcup_{\sigma\in H}\sigma\cdot
A^{\prime}_{H}\subseteq\bigcup_{\sigma\in H}\sigma\cdot
A_{H}=\left(\bigcap_{\sigma\in H}\sigma\cdot A\right)\cap X_{H}\subseteq A.$

Item (ii) is also straightforward:

$\displaystyle B^{\prime}$ $\displaystyle=\bigcup_{\sigma\in H}\sigma\cdot
B^{\prime}_{H}\subseteq\bigcup_{\sigma\in
H}\sigma\cdot\Bigl{(}\bigl{(}(A^{\prime}_{H}\times Y)\cap B_{H}\bigr{)}\times
Z\Bigr{)}$ $\displaystyle=\left(\left(\left(\bigcup_{\sigma\in H}\sigma\cdot
A^{\prime}_{H}\right)\times Y\right)\cap B_{H}\right)\times Z$
$\displaystyle=\left((A^{\prime}\times Y)\cap\left(\bigcap_{\sigma\in
H}\sigma\cdot B\right)\cap(X_{H}\times Y)\right)\times Z$
$\displaystyle\subseteq\bigl{(}(A^{\prime}\times Y)\cap B\bigr{)}\times Z.$

Item (iii) is straightforward too:

$\displaystyle\operatorname{dom}(G)$ $\displaystyle=\bigcup_{\sigma\in
H}\sigma\cdot\operatorname{dom}(G_{H})\supseteq\bigcup_{\sigma\in
H}\sigma\cdot B^{\prime}_{H}=B^{\prime}.$

For item (iv), if $(x,y,z)\in B^{\prime}$, then we have
$\tau_{x}\cdot(x,y,z)\in B^{\prime}_{H}$, so we get

$\displaystyle G^{\prime}\bigl{(}x,G(x,y,z)\bigr{)}$
$\displaystyle=\tau_{x}^{-1}\cdot\Biggl{(}G^{\prime}\biggl{(}\tau_{x}\cdot\Bigl{(}x,\tau_{x}^{-1}\cdot
G\bigl{(}\tau_{x}\cdot(x,y,z)\bigr{)}\Bigr{)}\biggr{)}\Biggr{)}$
$\displaystyle=\tau_{x}^{-1}\cdot\biggl{(}G^{\prime}\Bigl{(}\tau_{x}\cdot
x,G\bigl{(}\tau_{x}\cdot x,\tau_{x}\cdot(y,z)\bigr{)}\Bigr{)}\biggr{)}$
$\displaystyle=\tau_{x}^{-1}\cdot\bigl{(}\tau_{x}\cdot(y,z)\bigr{)}=(y,z).$

For item (v), if $(x,y,z)\in B^{\prime}$, then (since $\tau_{x}\cdot(x,y,z)\in
B^{\prime}_{H}$) we have

$\displaystyle\pi\bigl{(}G(x,y,z)\bigr{)}$
$\displaystyle=\pi\biggl{(}\tau_{x}^{-1}\cdot\Bigl{(}G\bigl{(}\tau_{x}\cdot(x,y,z)\bigr{)}\Bigr{)}\biggr{)}=\tau_{x}^{-1}\cdot\pi\Bigl{(}G\bigl{(}\tau_{x}\cdot(x,y),\tau_{x}\cdot
z\bigr{)}\Bigr{)}$ $\displaystyle=\tau_{x}^{-1}\cdot
g\bigl{(}\tau_{x}\cdot(x,y)\bigr{)}=g(x,y),$

where the last equality follows since $g$ is $H$-equivariant.

For item (vi), we have

$\displaystyle(\gamma[\mu]\otimes\nu)(B^{\prime})$
$\displaystyle=(\gamma[\mu]\otimes\nu)\left(\bigcup_{\sigma\in H}\sigma\cdot
B^{\prime}_{H}\right)=\lvert
H\rvert\cdot(\gamma[\mu]\otimes\nu)(B^{\prime}_{H})=(\gamma\mathord{|}_{X^{\prime}}[\mu_{H}]\otimes\nu)(B^{\prime}_{H})=1,$

where the second equality follows since $\gamma[\mu]$ is $H$-invariant, the
$H$-action on $Z$ is trivial and the orbits $(\sigma\cdot
X^{\prime})_{\sigma\in H}$ of $X^{\prime}$ are pairwise disjoint.

For item (vii), we have

$\displaystyle\mu(A^{\prime})$ $\displaystyle=\mu\left(\bigcup_{\sigma\in
H}\sigma\cdot A^{\prime}_{H}\right)=\lvert
H\rvert\cdot\mu(A^{\prime}_{H})=\mu_{H}(A^{\prime}_{H})=1,$

where the second equality follows since $\mu$ is $H$-invariant and
$A^{\prime}_{H}\subseteq A_{H}\subseteq X^{\prime}$ and the orbits
$(\sigma\cdot X^{\prime})_{\sigma\in H}$ of $X^{\prime}$ are pairwise
disjoint.

Finally, for item (viii)
($G(x,\mathord{-},\mathord{-})_{*}(\gamma(x)\otimes\nu)=\Theta(f(x))$ whenever
$x\in A^{\prime}$), we need to show that if $x\in A^{\prime}$ and
$(\bm{y},\bm{z})\sim(\gamma(x)\otimes\nu)$ then
$G(x,\bm{y},\bm{z})\sim\Theta(f(x))$.

Indeed, first note that since $\gamma$ is $H$-equivariant and $\nu$ is
$H$-invariant (as the $H$-action on $Z$ is trivial), it follows that

$\displaystyle\tau_{x}\cdot(\bm{y},\bm{z})$
$\displaystyle\sim\tau_{x}\cdot(\gamma(x)\otimes\nu)=\gamma(\tau_{x}\cdot
x)\otimes\nu$

so we get

$\displaystyle G(x,\bm{y},\bm{z})$ $\displaystyle=\tau_{x}^{-1}\cdot
G_{H}\bigl{(}\tau_{x}\cdot(x,\bm{y},\bm{z})\bigr{)}\sim\tau_{x}^{-1}\cdot\Theta\bigl{(}f\mathord{|}_{X^{\prime}}(\tau_{x}\cdot
x)\bigr{)}$ $\displaystyle=\tau_{x}^{-1}\cdot(\theta\bigl{(}f(\tau_{x}\cdot
x)\bigr{)}\otimes\eta)=\theta(f(x))\otimes\eta=\Theta(f(x)),$

where the distributional equality follows from item (viii) applied to
$(G_{H},\gamma\mathord{|}_{X^{\prime}})$ as $\tau_{x}\cdot x\in X^{\prime}$,
the second equality follows from the definition of $\Theta$, the third
equality follows since $f$ and $\theta$ are $H$-equivariant and $\eta$ is
$H$-invariant (as the $H$-action on $W$ is trivial). ∎

##  9 Amalgamating types

In this section we finally define amalgamating types and prove its
amalgamation property. Before we give the definition, let us point out that
one more technical detail was not covered in Section 8. Recall that the whole
purpose of sampling via amalgamating types is that we want
$(\ell-1)$-independent $d$-Euclidean structures ${\mathcal{N}}$ satisfying
$\texttt{UCouple}[\ell]$ to have a unique amalgamating $\ell$-type. We have
seen in Proposition 5.8 that such ${\mathcal{N}}$ have a unique overlapping
$\ell$-type, which in particular implies that they also have a unique
dissociated $\ell$-type; since our intuition is that amalgamating $\ell$-types
carry an intermediate amount of information between their dissociated and
overlapping counterparts, we would expect that they are also unique. However,
since to make Lemma 8.3 go through we had encoded (potential) extra
information within the set $[0,1]$, it is impossible to get uniqueness from
the output of Lemma 8.3.

We solve this issue by being pragmatic again: since Proposition 5.8 says that
an $(\ell-1)$-independent $d$-Euclidean structure ${\mathcal{N}}$ satisfying
$\texttt{UCouple}[\ell]$ has a unique overlapping $\ell$-type, we can define
amalgamating $A$-types in ${\mathcal{N}}$ by splitting into cases: if $\lvert
A\rvert\leq\ell$, then we simply let the amalgamating $A$-type be the
overlapping $A$-type and if $\lvert A\rvert>\ell$, we actually apply Lemma 8.3
(in both cases, we also add information about lower amalgamating types). Since
we also have to keep track of the additional countable set $V$ of the
dissociated and overlapping types and the same ${\mathcal{N}}$ can satisfy
$(\ell-1)$-independence and $\texttt{UCouple}[\ell]$ for multiple different
$\ell$, we call this construction the amalgamating $(A,V,\ell)$-type.

Finally, we point out that since Lemma 8.3 requires the addition of dummy
variables, our definition of $(A,V,\ell)$-type requires us to go from
${\mathcal{N}}$ to an auxiliary $\widehat{{\mathcal{N}}}$ that has extra dummy
variables.

Before we formalize this construction, let us introduce some terminology
regarding partial functions: we say that a diagram of partial functions is
_weakly commutative_ if for any two composition paths
$(\alpha_{1},\ldots,\alpha_{n})$ and $(\beta_{1},\ldots,\beta_{m})$ between
from the same $A$ to the same $B$, we have

$\displaystyle(\alpha_{n}\mathbin{\circ}\cdots\mathbin{\circ}\alpha_{1})\mathord{|}_{D}$
$\displaystyle=(\beta_{m}\mathbin{\circ}\cdots\mathbin{\circ}\beta_{1})_{D},$

where

$\displaystyle D$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\operatorname{dom}(\alpha_{n}\mathbin{\circ}\cdots\mathbin{\circ}\alpha_{1})\cap\operatorname{dom}(\beta_{m}\mathbin{\circ}\cdots\mathbin{\circ}\beta_{1})_{D},$

that is, the two compositions match in their common domain (but might not
match as partial functions because of points in which only one of them is
defined); weak equivariance is defined similarly. Since in all of our weakly
commutative diagrams the partial functions will be defined almost everywhere,
these can be thought of “almost everywhere commutative diagrams”.

We will also need the following invariant/equivariant version of Lemma 7.3:

###### Lemma 9.1.

Let $\Omega_{X}$, $\Omega_{Y}$ and $\Omega_{Z}$ be Borel spaces, let $\bm{X}$,
$\bm{Y}$ and $\bm{Z}$ be random variables with values in $\Omega_{X}$,
$\Omega_{Y}$ and $\Omega_{Z}$, respectively. Suppose also that $H$ is a
countable group acting measurably on $\Omega_{X}$, $\Omega_{Y}$ and
$\Omega_{Z}$ and that the distribution $\mu_{Z}$ of $\bm{Z}$ is $H$-invariant.
Then the following are equivalent:

  1. i.

$\bm{X}$ is conditionally independent from $\bm{Y}$ given $\bm{Z}$ and the
distribution of $(\bm{X},\bm{Z})$ is $H$-invariant.

  2. ii.

There exists an $H$-equivariant partial Markov kernel
$\rho\colon\Omega_{Z}\rightharpoonup{\mathcal{P}}(\Omega_{X})$ with
$\mu_{Z}(\operatorname{dom}(\rho))=1$ such that if $\bm{\widetilde{X}}$ is
picked at random according to $\rho(\bm{Z})$ conditionally on $\bm{Z}$ and
conditionally independently from $\bm{Y}$, then
$(\bm{\widetilde{X}},\bm{Y},\bm{Z})\sim(\bm{X},\bm{Y},\bm{Z})$.

###### Proof.

The implication (ii)$\implies$(i) is obvious as $\bm{\widetilde{X}}$ and
$\bm{Y}$ are conditionally independent given $\bm{Z}$ and the facts that
$\rho$ is $H$-equivariant and that $\mu_{Z}$ is $H$-invariant imply that the
distribution of $(\bm{\widetilde{X}},\bm{Z})$ is $H$-invariant.

For the implication (i)$\implies$(ii), we apply Lemma 7.3 to obtain a (not
necessarily $H$-equivariant) Markov kernel
$\rho^{\prime}\colon\Omega_{Z}\to{\mathcal{P}}(\Omega_{X})$ such that if
$\bm{\widetilde{X}}$ is picked at random according to $\rho^{\prime}(\bm{Z})$
conditionally on $\bm{Z}$ and conditionally independently from $\bm{Y}$, then
$(\bm{\widetilde{X}},\bm{Y},\bm{Z})\sim(\bm{X},\bm{Y},\bm{Z})$.

We now argue that $\rho^{\prime}$ is $\mu_{Z}$-almost everywhere
$H$-equivariant, that is, we argue that for $\mu_{Z}$-almost every
$z\in\Omega_{Z}$ and every $\tau\in H$, we have $\rho^{\prime}(\tau\cdot
z)=\tau\cdot\rho^{\prime}(z)$. For this, it suffices to show that for every
measurable $A\subseteq\Omega_{Z}$ and every measurable $B\subseteq\Omega_{X}$,
we have

$\displaystyle\int_{A}\rho^{\prime}(\tau\cdot z)(B)\ d\mu_{Z}(z)$
$\displaystyle=\int_{A}(\tau\cdot\rho^{\prime}(z))(B)\ d\mu_{Z}(z).$

But indeed, note that since $\mu_{Z}$ is $H$-invariant, we have

$\displaystyle\int_{A}\rho^{\prime}(\tau\cdot z)(B)\ d\mu_{Z}(z)$
$\displaystyle=\int_{\tau^{-1}(A)}\rho^{\prime}(z)(B)\
d\mu_{Z}(z)=\mathbb{P}[\bm{Z}\in\tau^{-1}(A),\bm{\widetilde{X}}\in B]$
$\displaystyle=\mathbb{P}[\bm{Z}\in
A,\bm{\widetilde{X}}\in\tau(B)]=\int_{A}\rho^{\prime}(z)(\tau(B))\
d\mu_{Z}(z)=\int_{A}(\tau\cdot\rho^{\prime}(z))(B)\ d\mu_{Z}(z),$

where the third equality follows since
$(\bm{\widetilde{X}},\bm{Z})\sim(\bm{X},\bm{Z})$ and this latter distribution
is $H$-invariant.

Thus $\rho^{\prime}$ is $\mu_{Z}$-almost everywhere $H$-equivariant. Since $H$
is countable, this means that the set

$\displaystyle D$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\\{z\in\Omega_{Z}\mid\forall\tau\in
H,\rho^{\prime}(\tau\cdot z)=\tau\cdot\rho^{\prime}(z)\\}$

has $\mu_{Z}$-measure $1$, so letting
$\rho\colon\Omega_{Z}\rightharpoonup{\mathcal{P}}(\Omega_{X})$ be the partial
Markov kernel defined to be equal to $\rho$ on $D$ and not defined anywhere
else yields the result. ∎

We are now finally ready to define amalgamating types.

###### Definition 9.2.

Let $A$ be a finite set, let $V$ be a countably infinite131313Differently from
dissociated or overlapping types, amalgamating types do not make sense when
$V$ is finite. This is because for the definition to go through (see Lemma 9.5
below), we make use of Proposition 7.4, which requires $V$ to be infinite. set
disjoint from $A$ and let $\ell\in\mathbb{N}$. Let also $\Omega=(X,\mu)$ be an
atomless standard probability space and $d\in\mathbb{N}$. The space of
_amalgamating $(A,V,\ell)$-types_ is defined inductively as

$\displaystyle{\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\begin{dcases*}\left(\prod_{B\in
r(A,\lvert
A\rvert-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg}}(\Omega)\right)\times{\mathcal{S}}_{A,V}^{(d),\operatorname{ovlp}}(\Omega),&if
$\lvert A\rvert\leq\ell$,\\\ \left(\prod_{B\in r(A,\lvert
A\rvert-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg}}(\Omega)\right)\times{\mathcal{S}}_{A,V}^{(d),\operatorname{dsct}}(\Omega)\times[0,1],&if
$\lvert A\rvert>\ell$,\end{dcases*}$

equipped with the product topology (which makes it a Polish space, hence a
Borel space, see Proposition 6.6(ii)). We think of
${\mathcal{S}}_{A,V}^{(d),\operatorname{ovlp}}(\Omega)$ and
${\mathcal{S}}_{A,V}^{(d),\operatorname{dsct}}(\Omega)\times[0,1]$ in the
above as the coordinate indexed by $A$.

We also define
$\pi^{\operatorname{amlg},\ell}_{A,V}\colon{\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}\to{\mathcal{S}}_{A,V}^{(d),\operatorname{dsct}}$
by letting it be the natural projection when $\lvert A\rvert>\ell$ and letting
it be the composition of the natural projection onto
${\mathcal{S}}_{A,V}^{(d),\operatorname{ovlp}}$ with
$\pi^{\operatorname{ovlp}}_{A,V}\colon{\mathcal{S}}_{A,V}^{(d),\operatorname{ovlp}}\to{\mathcal{S}}_{A,V}^{(d),\operatorname{dsct}}$
when $\lvert A\rvert\leq\ell$.

We further define
$M^{\operatorname{amlg},\ell}_{A,V}\colon{\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}\to{\mathcal{K}}_{A}$
as the composition
$M^{\operatorname{amlg},\ell}_{A,V}\stackrel{{\scriptstyle\text{def}}}{{=}}M^{\operatorname{dsct}}_{A,V}\mathbin{\circ}\pi^{\operatorname{amlg},\ell}_{A,V}$;
as before, for an amalgamating type
$p\in{\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}$, we call
$M^{\operatorname{amlg},\ell}_{A,V}(p)$ the _underlying model_ of $p$.

For an injection $\alpha\colon A\to B$ with $B$ disjoint from $V$ we define a
map
$\alpha^{*}\colon{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}\to{\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}$
inductively on the size of $A$ by

$\displaystyle\alpha^{*}(p)_{C}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\begin{dcases*}\alpha\mathord{\downarrow}_{C}^{*}(p_{\alpha(C)}),&if
$\lvert C\rvert<\lvert A\rvert$,\\\ \alpha^{*}(p_{B}),&if $C=A$ and $\lvert
A\rvert=\lvert B\rvert$,\\\
\alpha\mathord{\downarrow}_{A}^{*}((p_{\alpha(A)})_{\alpha(A)}),&if $C=A$ and
$\lvert A\rvert<\lvert B\rvert$,\end{dcases*}$

for every $p\in{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}$ and every
$C\in r(A)$, where in the second case, $\alpha^{*}$ is either the map
${\mathcal{S}}_{B,V}^{(d),\operatorname{ovlp}}\to{\mathcal{S}}_{A,V}^{(d),\operatorname{ovlp}}$
defined contra-variantly in Definition 5.4 when $\lvert B\rvert\leq\ell$ or
the extension of the map
${\mathcal{S}}_{B,V}^{(d),\operatorname{dsct}}\to{\mathcal{S}}_{A,V}^{(d),\operatorname{dsct}}$
to a map
${\mathcal{S}}_{B,V}^{(d),\operatorname{dsct}}\times[0,1]\to{\mathcal{S}}_{A,V}^{(d),\operatorname{dsct}}\times[0,1]$
by acting identically on the second coordinate when $\lvert B\rvert>\ell$; in
the third case, $\alpha\mathord{\downarrow}_{A}^{*}$ is defined similarly and
we take the $\alpha(A)$th coordinate twice, the first time corresponds to the
projection
${\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}\to{\mathcal{S}}_{\alpha(A),V}^{(d),\operatorname{amlg},\ell}$
and the second time corresponds to the projection of
${\mathcal{S}}_{\alpha(A),V}^{(d),\operatorname{amlg},\ell}$ onto either
${\mathcal{S}}_{\alpha(A),V}^{(d),\operatorname{ovlp}}$ when $\lvert
A\rvert\leq\ell$ or
${\mathcal{S}}_{\alpha(A),V}^{(d),\operatorname{dsct}}\times[0,1]$ when
$\lvert A\rvert>\ell$.

It is straightforward to check that the fact that the maps of Definition 5.4
are contra-variant (see Lemma 5.5) implies that the maps defined above are
also contra-variant.

In turn, recall that this also contra-variantly induces maps

$\displaystyle\alpha^{*}\colon\prod_{C\in r(B,\lvert
B\rvert-1)}{\mathcal{S}}_{C,V}^{(d),\operatorname{amlg},\ell}\to\prod_{C\in
r(A,\lvert A\rvert-1)}{\mathcal{S}}_{C,V}^{(d),\operatorname{amlg},\ell}$

given by

$\displaystyle\alpha^{*}(p)_{C}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(\alpha\mathord{\downarrow}_{C})^{*}(p_{\alpha(C)})\qquad(p\in{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell},C\in
r(A,\lvert A\rvert-1)).$

Finally, it is obvious that the maps $\pi^{\operatorname{amlg},\ell}$ and
$M^{\operatorname{amlg},\ell}$ are equivariant, that is, we have

$\displaystyle\pi^{\operatorname{amlg},\ell}_{A,V}\mathbin{\circ}\alpha^{*}$
$\displaystyle=\alpha^{*}\mathbin{\circ}\pi^{\operatorname{amlg},\ell}_{B,V},$
$\displaystyle M^{\operatorname{amlg},\ell}_{A,V}\mathbin{\circ}\alpha^{*}$
$\displaystyle=\alpha^{*}\mathbin{\circ}M^{\operatorname{amlg},\ell}_{B,V},$
(9.1)

Let now ${\mathcal{N}}$ be a $d$-Euclidean structure in ${\mathcal{L}}$ over
$\Omega$ and let $\widehat{{\mathcal{N}}}$ be the $d$-Euclidean structure in
${\mathcal{L}}$ over $\Omega^{2}$ given by

$\displaystyle\widehat{{\mathcal{N}}}_{P}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\\{(x,x^{\prime})\in{\mathcal{E}}_{k(P)}^{(d)}(\Omega)\times{\mathcal{E}}_{k(P)}(\Omega)\mid
x\in{\mathcal{N}}_{P}\\}\qquad(P\in{\mathcal{L}}).$ (9.2)

(In the above, all order variables are contained within $x$.)

Clearly $\phi_{\mathcal{N}}=\phi_{\widehat{{\mathcal{N}}}}$. In fact, note
that for every
$(x,x^{\prime})\in{\mathcal{E}}_{A}^{(d)}(\Omega)\times{\mathcal{E}}_{A}(\Omega)$
we have

$\displaystyle\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{A,V}(x,x^{\prime})(y,y^{\prime})$
$\displaystyle=\mathbbm{p}^{\operatorname{dsct},{\mathcal{N}}}_{A,V}(x)(y)\qquad((y,y^{\prime})\in{\mathcal{E}}_{A,V}^{(d),\operatorname{dsct}}(\Omega)\times{\mathcal{E}}_{A,V}^{(0),\operatorname{dsct}}(\Omega)),$
(9.3)
$\displaystyle\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{A,V}(x,x^{\prime})(y,y^{\prime})$
$\displaystyle=\mathbbm{p}^{\operatorname{ovlp},{\mathcal{N}}}_{A,V}(x)(y)\qquad((y,y^{\prime})\in{\mathcal{E}}_{A,V}^{(d),\operatorname{ovlp}}(\Omega)\times{\mathcal{E}}_{A,V}^{(0),\operatorname{ovlp}}(\Omega)).$

Suppose now that ${\mathcal{N}}$ is $(\ell-1)$-independent and satisfies
$\texttt{UCouple}[\ell]$ and note that $\widehat{{\mathcal{N}}}$ inherits the
same properties. We will define
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$,
$\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$ and
$\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$ by induction
on the size of $A$ so that as to satisfy the following properties:

  1. a.

$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}\colon{\mathcal{E}}_{A}^{(d)}(\Omega^{2})\rightharpoonup{\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$
is a measurable partial function that is defined $\mu^{(d)}$-almost
everywhere. Furthermore, if $\lvert A\rvert\leq\ell$, then
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}$ is constant
on its domain.
($\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$ gives
the amalgamating $(A,V,\ell)$-type of
$x\in{\mathcal{E}}_{A}^{(d)}(\Omega^{2})$.)

  2. b.

When the amalgamating $(A,V,\ell)$-type is defined, then all lower
amalgamating types are defined: if
$x\in\operatorname{dom}(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V})$,
then for every $B\in r(A,\lvert A\rvert-1)$, we have
$\iota_{B,A}^{*}(x)\in\operatorname{dom}(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V})$
(recall that $\iota_{B,A}\colon B\to A$ is the inclusion map).

  3. c.

$\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}\colon{\mathcal{E}}_{A,\lvert
A\rvert-1}^{(d)}(\Omega^{2})\times{\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})\rightharpoonup{\mathcal{S}}_{A,V}^{(d),\operatorname{ovlp}}(\Omega^{2})$
is a measurable partial function such that for every
$x\in\operatorname{dom}(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V})$,
we have

$\displaystyle\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}\bigl{(}\tau_{A}(x),\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(x)\bigr{)}$
$\displaystyle=\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{A,V}(x),$

where
$\tau_{A}\colon{\mathcal{E}}_{A}^{(d)}(\Omega^{2})\to{\mathcal{E}}_{A,\lvert
A\rvert-1}^{(d)}(\Omega^{2})$ is the natural projection
($\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$ is the
function that retrieves the overlapping type from the amalgamating type along
with lower pointwise data).

  4. d.

$\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}\colon\prod_{B\in
r(A,\lvert
A\rvert-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})\rightharpoonup{\mathcal{P}}({\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2}))$
is a partial Markov kernel that is $S_{A}$-equivariant and is such that for
the natural projection
$\tau_{A}\colon{\mathcal{E}}_{A}^{(d)}(\Omega^{2})\to{\mathcal{E}}_{A,\lvert
A\rvert-1}^{(d)}(\Omega^{2})$, if $\bm{x}$ is sampled in
${\mathcal{E}}_{A}^{(d)}(\Omega^{2})$ according to $\mu^{(d)}$, then the
conditional distribution of
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(\bm{x})$
given $\tau_{A}(\bm{x})$ is

$\displaystyle\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}\Bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V}\bigl{(}\iota_{B,A}^{*}(\bm{x})\bigr{)}\mathrel{\Big{|}}B\in
r(A,\lvert A\rvert-1)\Bigr{)}.$

In particular,
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(\bm{x})$
is conditionally independent from $\tau_{A}(\bm{x})$ given
$(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V}(\iota_{B,A}^{*}(\bm{x}))\mid
B\in r(A,\lvert A\rvert-1))$
($\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$ then gives
how to sample the amalgamating $(A,V,\ell)$-type from the lower amalgamating
types).

Furthermore, if $\lvert A\rvert\leq\ell$, then
$\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$ is the
constant function that maps every point to the distribution concentrated on
the unique point in the image of
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$ (see
item (a) above).

  5. e.

For every injection $\beta\colon A^{\prime}\to A$ with $A^{\prime}$ also
disjoint from $V$, the diagram

${{\mathcal{E}}_{A}^{(d)}(\Omega^{2})}$${{\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})}$${{\mathcal{S}}_{A,V}^{(d),\operatorname{dsct},\ell}(\Omega^{2})}$${{\mathcal{K}}_{A}}$${{\mathcal{E}}_{A^{\prime}}^{(d)}(\Omega^{2})}$${{\mathcal{S}}_{A^{\prime},V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})}$${{\mathcal{S}}_{A^{\prime},V}^{(d),\operatorname{dsct},\ell}(\Omega^{2})}$${{\mathcal{K}}_{A^{\prime}}}$$\scriptstyle{\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}}$$\scriptstyle{\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{A,V}}$$\scriptstyle{M^{\widehat{{\mathcal{N}}}}_{A}}$$\scriptstyle{\beta^{*}}$$\scriptstyle{\pi^{\operatorname{amlg},\ell}_{A,V}}$$\scriptstyle{M^{\operatorname{amlg},\ell}_{A,V}}$$\scriptstyle{\beta^{*}}$$\scriptstyle{M^{\operatorname{dsct}}_{A,V}}$$\scriptstyle{\beta^{*}}$$\scriptstyle{\beta^{*}}$$\scriptstyle{\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}}$$\scriptstyle{\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{A^{\prime},V}}$$\scriptstyle{M^{\widehat{{\mathcal{N}}}}_{A^{\prime}}}$$\scriptstyle{\pi^{\operatorname{amlg},\ell}_{A^{\prime},V}}$$\scriptstyle{M^{\operatorname{amlg},\ell}_{A^{\prime},V}}$$\scriptstyle{M^{\operatorname{dsct}}_{A^{\prime},V}}$

is weakly commutative.

The final property is weak amalgamation, which is a joint property of all sets
of size $\lvert A\rvert$:

  1. vi.

Weak amalgamation: if $B$ is a set with $\lvert B\rvert=\lvert A\rvert+1$ and
$B\cap V=\varnothing$ and $\bm{x}$ is sampled in
${\mathcal{E}}_{B}^{(d)}(\Omega^{2})$ according to $\mu^{(d)}$, then
$\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{B,V}(\bm{x})$ is
conditionally independent from $\tau_{B}(\bm{x})$ given

$\displaystyle\bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}(\iota_{A^{\prime},B}(\bm{x}))\mathrel{\big{|}}A^{\prime}\in
r(B,\lvert B\rvert-1)\bigr{)},$

where
$\tau_{B}\colon{\mathcal{E}}_{B}^{(d)}(\Omega^{2})\to{\mathcal{E}}_{B,\lvert
B\rvert-1}^{(d)}(\Omega^{2})$ is the natural projection.

In this definition, we focus on describing the construction, deferring the
proof of some of the claimed properties to Lemmas 9.4 and 9.5.

First we consider the case when $\lvert A\rvert\leq\ell$. Since
$\widehat{{\mathcal{N}}}$ is $(\ell-1)$-independent and satisfies
$\texttt{UCouple}[\ell]$, Proposition 5.8 says that there exists an
overlapping $(A,V)$-type
$q_{A}\in{\mathcal{S}}_{A,V}^{(d),\operatorname{ovlp}}(\Omega^{2})$ such that
for $\mu^{(d)}$-almost every $x\in{\mathcal{E}}_{A}^{(d)}(\Omega^{2})$, we
have
$\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{A,V}(x)=q_{A}$.
Note also that by inductive hypothesis, we know that for every $B\in
r(A,\lvert A\rvert-1)$, for almost every
$x\in{\mathcal{E}}_{A}^{(d)}(\Omega^{2})$, we have
$\iota_{B,A}^{*}(x)\in\operatorname{dom}(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V})$.
Thus, the set

$\displaystyle
D\stackrel{{\scriptstyle\text{def}}}{{=}}\\{x\in{\mathcal{E}}_{A}^{(d)}(\Omega^{2})$
$\displaystyle\mid\forall\beta\in
S_{A},\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{A,V}(\beta^{*}(x))=q_{A}$
$\displaystyle\land\forall B\in r(A,\lvert
A\rvert-1),\iota_{B}^{*}(x)\in\operatorname{dom}(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V})\\}$

has $\mu^{(d)}$-measure $1$, so we define

$\displaystyle\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(x)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\biggl{(}\Bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V}\bigl{(}\iota_{B,A}^{*}(x)\bigr{)}\mathrel{\Big{|}}B\in
r(A,\lvert A\rvert-1)\Bigr{)},q_{A}\biggr{)}$

whenever $x\in D$ and leave
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(x)$
undefined when $x\notin D$.

Since inductively all
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V}$ for
$B\in r(A,\lvert A\rvert-1)$ are constant on their domains, it follows that
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$ is also
constant on its domain $D$ (i.e., item (a) holds).

By definition it is also clear that item (b) holds (i.e., if
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$ is
defined, then all lower amalgamating types are also defined).

We now define
$\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}\colon{\mathcal{E}}_{A,\lvert
A\rvert-1}^{(d)}(\Omega^{2})\times{\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})\rightharpoonup{\mathcal{S}}_{A,V}^{(d),\operatorname{ovlp}}(\Omega^{2})$
by

$\displaystyle\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(x,p)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\rho_{A}(p),\qquad(x\in{\mathcal{E}}_{A,\lvert
A\rvert-1}^{(d)}(\Omega^{2}),p\in{\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})),$

where
$\rho_{A}\colon{\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})\to{\mathcal{S}}_{A,V}^{(d),\operatorname{ovlp}}(\Omega^{2})$
is the natural projection.

Item (c)
($\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(\tau_{A}(x),\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(x))=\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{A,V}(x)$
whenever
$x\in\operatorname{dom}(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V})$)
follows immediately from definitions.

Finally, define the partial Markov kernel

$\displaystyle\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}\colon\prod_{B\in
r(A,\lvert
A\rvert-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$
$\displaystyle\rightharpoonup{\mathcal{P}}({\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$

to be the constant function whose value is the distribution on
${\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$ that is
concentrated on the unique point in the image of
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$.
Obviously, item (d) holds (i.e.,
$\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$ correctly
computes the conditional distribution of the amalgamating $(A,V,\ell)$-type
given the lower amalgamating types).

We defer the proofs of items (e) and (vi) to Lemmas 9.4 and 9.5, respectively.

We now consider the case when $A=[a]$ for some $a\in\mathbb{N}$ with $a>\ell$.
Our objective is to setup for applying Lemma 8.3. First, we take the following
choice of parameters for the spaces:

$\displaystyle(X,\mu)$
$\displaystyle\leftarrow({\mathcal{E}}_{a,a-1}^{(d)}(\Omega^{2}),\mu^{(d)}),$
$\displaystyle(Z,\nu)$ $\displaystyle\leftarrow(\Omega,\mu),$
$\displaystyle(W,\eta)$ $\displaystyle\leftarrow([0,1],\lambda),$
$\displaystyle Y$
$\displaystyle\leftarrow{\mathcal{S}}_{a,V}^{(d),\operatorname{ovlp}}(\Omega^{2}),$
$\displaystyle U$ $\displaystyle\leftarrow\prod_{B\in
r(a,a-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2}),$
$\displaystyle V$
$\displaystyle\leftarrow{\mathcal{S}}_{a,V}^{(d),\operatorname{dsct}}(\Omega^{2}).$

Note that when we consider any of the contra-variant definitions only for
members of $S_{a}$ (i.e., injections $\beta\colon[a]\to[a]$, which are
necessarily bijections), contra-variance translates into a right action of
$S_{a}$ on the corresponding space.

It is obvious that the action of $S_{a}$ on
${\mathcal{E}}_{a,a-1}^{(d)}(\Omega^{2})$ is measurable and $\mu^{(d)}$ is
$S_{a}$-invariant. We claim that there exists a measurable set
$X^{\prime}\subseteq{\mathcal{E}}_{a,a-1}^{(d)}(\Omega^{2})$ such that the
orbits $(\sigma\cdot X^{\prime})_{\sigma\in S_{a}}$ of $X^{\prime}$ under the
action of $S_{a}$ are pairwise disjoint and $\mu(\bigcup_{\sigma\in
S_{a}}\sigma\cdot X^{\prime})=1$. Indeed, if we fix a measurable order $\leq$
on $\Omega^{2}$ (this can be done by noting that $\Omega^{2}$ is Borel-
isomorphic to $[0,1]$ and pulling back the usual order of $[0,1]$ through the
Borel-isomorphism), then we can take

$\displaystyle X^{\prime}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\\{x\in{\mathcal{E}}_{a,a-1}^{(d)}(\Omega^{2})\mid
x_{\\{1\\}}<x_{\\{2\\}}<\cdots<x_{\\{a\\}}\\}.$

The fact that $\mu^{(d)}(\bigcup_{\sigma\in S_{a}}\sigma\cdot X^{\prime})=1$
follows since $\mu$ is atomless.

By Proposition 6.6(v), the actions of $S_{a}$ on
${\mathcal{S}}_{a,V}^{(d),\operatorname{ovlp}}(\Omega^{2})$ and
${\mathcal{S}}_{a,V}^{(d),\operatorname{dsct}}(\Omega^{2})$ are continuous,
hence measurable. Similarly, the same proposition (along with the fact that
${\mathcal{S}}_{a,V}^{(d),\operatorname{amlg}}(\Omega^{2})$ is equipped with
product topology) implies that the action of $S_{a}$ on $\prod_{B\in
r(a,a-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$ is
continuous, hence measurable.

Also, obviously all trivial actions are measurable. Thus, we have covered all
hypotheses of Lemma 8.3 concerning action measurability and invariance (but
not yet equivariance of $f$, $g$, $\gamma$ and $\theta$).

Let us now setup the final objects $f$, $g$, $\gamma$ and $\theta$ of Lemma
8.3.

Note that if $\bm{x}$ is picked at random in
${\mathcal{E}}_{a}^{(d)}(\Omega^{2})$ according to $\mu^{(d)}$, then Lemma 5.7
implies that the distribution of
$\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{a,V}(\bm{x})$ is
$S_{a}$-invariant (as $\mu^{(d)}$ is $S_{a}$-invariant). Since inductively all
lower sized amalgamating types satisfy the weak amalgamation property of item
(vi), Lemma 9.1 gives us an $S_{a}$-equivariant partial Markov kernel

$\displaystyle\theta_{a}\colon\prod_{B\in
r(a,a-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})\rightharpoonup{\mathcal{P}}({\mathcal{S}}_{a,V}^{(d),\operatorname{dsct}}(\Omega^{2}))$

such that the conditional distribution of
$\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{a,V}(\bm{x})$
given $\tau_{a}(\bm{x})$ is

$\displaystyle\theta_{a}\Bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V}\bigl{(}\iota_{B,[a]}^{*}(\bm{x})\bigr{)}\mathrel{\Big{|}}B\in
r(a,a-1)\Bigr{)}.$

On the other hand, Lemma 5.7 also implies that the distribution of
$\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{A,V}(\bm{x})$ is
$S_{a}$-invariant as well, so Lemma 9.1 (with $\bm{Y}$ trivial) further gives
us an $S_{a}$-equivariant partial Markov kernel

$\displaystyle\gamma_{a}\colon{\mathcal{E}}_{a,a-1}^{(d)}(\Omega^{2})\rightharpoonup{\mathcal{P}}({\mathcal{S}}_{a,V}^{(d),\operatorname{ovlp}}(\Omega^{2}))$

such the conditional distribution of
$\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{a,V}(\bm{x})$
given $\tau_{a}(\bm{x})$ is $\gamma_{a}(\tau_{a}(\bm{x}))$ (in particular, we
have $\mu^{(d)}(\operatorname{dom}(\gamma_{a}))=1$).

We then take the following parameters for Lemma 8.3:

$\displaystyle\theta$ $\displaystyle\leftarrow\theta_{a},$
$\displaystyle\gamma$ $\displaystyle\leftarrow\gamma_{a},$

we define the partial functions

$\displaystyle f\colon{\mathcal{E}}_{a,a-1}^{(d)}(\Omega^{2})$
$\displaystyle\rightharpoonup\prod_{B\in
r(a,a-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2}),$
$\displaystyle
g\colon{\mathcal{E}}_{a,a-1}^{(d)}(\Omega^{2})\times{\mathcal{S}}_{a,V}^{(d),\operatorname{ovlp}}(\Omega^{2})$
$\displaystyle\rightharpoonup{\mathcal{S}}_{a,V}^{(d),\operatorname{dsct}}(\Omega^{2})$

by

$\displaystyle f(x)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V}(\iota_{B,[a]}^{*}(x))\mid
B\in r(a,a-1)),$ $\displaystyle g(x,p)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\pi^{\operatorname{ovlp}}_{a,V}(p),$
(9.4)

for every $x\in{\mathcal{E}}_{a,a-1}^{(d)}(\Omega^{2})$ and every
$p\in{\mathcal{S}}_{a,V}^{(d),\operatorname{ovlp}}(\Omega^{2})$ and we set

$\displaystyle A$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\operatorname{dom}(f)\cap
f^{-1}(\operatorname{dom}(\theta_{a}))\cap\operatorname{dom}(\gamma_{a}),$
$\displaystyle B$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\operatorname{dom}(g)={\mathcal{E}}_{a,a-1}^{(d)}(\Omega^{2})\times{\mathcal{S}}_{a,V}^{(d),\operatorname{ovlp}}(\Omega^{2})$

($g$ is actually a total function). Our inductive hypothesis of item (a)
implies that $\mu^{(d)}(A)=\gamma_{a}[\mu^{(d)}](B)=1$.

Let us now check that for every $x\in A$, we have

$\displaystyle g(x,\mathord{-})_{*}(\gamma_{a}(x))$
$\displaystyle=\theta_{a}(f(x)).$

Since $\mu^{(d)}(A)=1$ and $A\subseteq
f^{-1}(\operatorname{dom}(\theta_{a}))\cap\operatorname{dom}(\gamma_{a})$,
this is of course equivalent to showing that for $\bm{x}$ picked at random
according to $\mu^{(d)}$, we have

$\displaystyle g(\bm{x},\mathord{-})_{*}(\gamma_{a}(\bm{x}))$
$\displaystyle=\theta_{a}(f(\bm{x})).$

By construction, the right-hand side of the above is the conditional
distribution of
$\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{a,V}(\bm{x})$
given $\tau_{a}(\bm{x})$ and the left-hand side is the conditional
distribution of
$\pi^{\operatorname{ovlp}}_{a,V}(\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{a,V}(\bm{x}))$
given $\tau_{a}(\bm{x})$, so their equality follows from
$\pi^{\operatorname{ovlp}}_{a,V}\mathbin{\circ}\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{a,V}=\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{a,V}$
of Lemma 5.7.

Note also that Lemma 5.7 implies that $g$ is $S_{a}$-equivariant and our
inductive hypothesis of item (e) implies that $f$ is $S_{a}$-equivariant.

Thus, all hypotheses of Lemma 8.3 are satisfied. Let then
$(G,G^{\prime},A^{\prime},B^{\prime},\Theta)$ be given by the lemma and define

$\displaystyle\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}\colon{\mathcal{E}}_{a}^{(d)}(\Omega^{2})$
$\displaystyle\rightharpoonup{\mathcal{S}}_{a,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2}),$
$\displaystyle\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}\colon{\mathcal{E}}_{a,a-1}^{(d)}(\Omega^{2})\times{\mathcal{S}}_{a,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$
$\displaystyle\rightharpoonup{\mathcal{S}}_{a,V}^{(d),\operatorname{ovlp}}(\Omega^{2}),$
$\displaystyle\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}\colon\prod_{B\in
r(a,a-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$
$\displaystyle\rightharpoonup{\mathcal{P}}({\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2}))$

by

$\displaystyle\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}(x,(y,z))$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\biggl{(}\Bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V}\bigl{(}\iota_{B,[a]}^{*}(x)\bigr{)}\mid
B\in
r(a,a-1)\Bigr{)},G\Bigl{(}x,\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{a,V}\bigl{(}x,(y,z)\bigr{)},z\Bigr{)}\biggr{)}$

for every $x\in{\mathcal{E}}_{a,a-1}^{(d)}(\Omega^{2})$, every
$y\in\Omega\times{\mathcal{O}}_{A}^{d}$ and every $z\in\Omega$ (so that
$(x,y,z)\in{\mathcal{E}}_{a}^{(d)}(\Omega^{2})$, with $(y,z)$ containing the
coordinate indexed by $A$) such that

$\displaystyle\Bigr{(}x,\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{a,V}\bigl{(}x,(y,z)\bigr{)},z\Bigr{)}$
$\displaystyle\in B^{\prime},$

otherwise, we leave
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}(x,(y,z))$
undefined;

$\displaystyle\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}(x,p)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\rho^{\prime}_{a}(G^{\prime}(x,\rho_{a}(p))),$

for every $x\in{\mathcal{E}}_{a,a-1}^{(d)}(\Omega^{2})$ and every
$p\in{\mathcal{S}}_{a,a-1}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$, where
$\rho_{a}\colon{\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})\to{\mathcal{S}}_{A,V}^{(d),\operatorname{dsct}}(\Omega^{2})\times[0,1]$
and
$\rho^{\prime}_{a}\colon{\mathcal{S}}_{A,V}^{(d),\operatorname{ovlp}}(\Omega^{2})\times\Omega\to{\mathcal{S}}_{A,V}^{(d),\operatorname{ovlp}}(\Omega^{2})$
are the natural projections; and

$\displaystyle\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}(q)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\delta_{q}\otimes\Theta(q),$

for every $q\in\prod_{B\in
r(a,a-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$,
where $\delta_{q}$ is the Dirac measure concentrated on $q$.

We now argue why Lemma 8.3 gives all required properties to continue the
inductive construction, with the exception of properties (e) and (vi) whose
proofs are deferred to Lemmas 9.4 and 9.5, respectively.

Since $G$ is measurable and is defined $\gamma_{a}[\mu^{(d)}]$-almost
everywhere (by items (iii) and (vi) of Lemma 8.3), it follows that
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}$ is measurable
and is defined almost everywhere, so item (a) holds.

Items (i) and (ii) of Lemma 8.3 imply that when the amalgamating
$(a,V,\ell)$-type is defined, then all lower amalgamating types are also
defined, that is, item (b) holds.

The fact that
$\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}$
correctly retrieves the overlapping type from the amalgamating type along with
lower pointwise data (item (c)) follows from item (iv) in Lemma 8.3: for every
$(x,y,z)\in\operatorname{dom}(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V})$
with $x\in{\mathcal{E}}_{a,a-1}^{(d)}(\Omega)$,
$y\in\Omega\times{\mathcal{O}}_{a}^{d}$ and $z\in\Omega$, we have

$\displaystyle\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}\Bigl{(}x,\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}\bigl{(}x,(y,z)\bigr{)}\Bigr{)}$
$\displaystyle=\rho^{\prime}_{a}\Biggl{(}G^{\prime}\biggl{(}x,G\Bigl{(}x,\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{a,V}\bigl{(}x,(y,z)\bigr{)},z\Bigr{)}\biggr{)}\Biggr{)}$
$\displaystyle=\rho^{\prime}_{a}\Bigl{(}\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{a,V}\bigl{(}x,(y,z)\bigr{)},z\Bigr{)}=\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{a,V}\bigl{(}x,(y,z)\bigr{)}.$

For item (d), we have to show that if $\bm{x}$ is sampled in
${\mathcal{E}}_{a}^{(d)}(\Omega^{2})$ according to $\mu^{(d)}$, then the
conditional distribution of
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}(\bm{x})$
given $\tau_{a}(\bm{x})$ is

$\displaystyle\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}\Bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V}\bigl{(}\iota_{B,[a]}^{*}(\bm{x})\bigr{)}\mathrel{\Big{|}}B\in
r(a,a-1)\Bigr{)}.$

Note that the argument of
$\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}$ is precisely
$f(\bm{x})$.

Since $\mu^{(d)}(A^{\prime})=1$, it suffices to show that for every $x\in
A^{\prime}\subseteq{\mathcal{E}}_{a,a-1}^{(d)}(\Omega^{2})$, if $\bm{y}$ and
$\bm{z}$ are picked independently at random in
$\Omega\times{\mathcal{O}}_{a}^{d}$ and $\Omega$ with respect to
$\mu\otimes\bigotimes_{i=1}^{d}\nu_{a}$ and $\mu$, respectively, then

$\displaystyle\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}(x,(\bm{y},\bm{z}))$
$\displaystyle\sim\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}\bigl{(}f(\bm{x})\bigr{)}.$

By the definition of
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}$ and
$\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}$, it is clear
that all coordinates corresponding to lower amalgamating types correctly
match, so the above is equivalent to

$\displaystyle
G\Bigl{(}x,\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{a,V}\bigl{(}x,(\bm{y},\bm{z})\bigr{)},\bm{z}\Bigr{)}$
$\displaystyle\sim\Theta\bigl{(}f(x)\bigr{)}.$

Recalling that $\bm{z}$ is a dummy variable for $\widehat{{\mathcal{N}}}$
(more specifically, using (9.3)), we note that
$\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{a,V}\bigl{(}x,(\bm{y},\bm{z})\bigr{)}$
is independent from $\bm{z}$, so their joint distribution is precisely
$\gamma_{a}(x)\otimes\mu$, so the above is equivalent to

$\displaystyle G(x,\mathord{-},\mathord{-})_{*}(\gamma_{a}(x)\otimes\mu)$
$\displaystyle=\Theta\bigl{(}f(x)\bigr{)},$

which is precisely item (viii) of Lemma 8.3.

Finally, we consider the case when $\lvert A\rvert>\ell$ and $A$ is not $[a]$
for some $a\in\mathbb{N}$. We fix a bijection $\alpha_{A}\colon[\lvert
A\rvert]\to A$ and define

$\displaystyle\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(\alpha_{A}^{-1})^{*}\mathbin{\circ}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{\lvert
A\rvert,V}\mathbin{\circ}\alpha_{A}^{*},$ (9.5)
$\displaystyle\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(\alpha_{A}^{-1})^{*}\mathbin{\circ}\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{\lvert
A\rvert,V}\mathbin{\circ}\alpha_{A}^{*},$
$\displaystyle\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(\alpha_{A}^{-1})^{*}\mathbin{\circ}\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{\lvert
A\rvert,V}\mathbin{\circ}\alpha_{A}^{*}.$

It will be convenient to let
$\alpha_{[a]}\stackrel{{\scriptstyle\text{def}}}{{=}}\operatorname{id}_{[a]}$
for every $a\in\mathbb{N}_{+}$ so that the equalities in the above remain true
even in the case $A=[a]$, in which we define the partial functions above
directly. It is also clear that the equalities above also hold when $\lvert
A\rvert\leq\ell$.

It is straightforward to check that the properties required by the
construction for $A$ follow as a consequence of the properties for $[\lvert
A\rvert]$, except for the two joint properties of items (e) and (vi) that are
being deferred.

Note also that once we show that item (e) holds in Lemma 9.4, a direct
consequence is that the definition above does not depend on the choice of
$\alpha_{A}$, except for a zero-measure set.

###### Remark 9.3.

A direct consequence of the item (a) of Definition 9.2 is that if
${\mathcal{N}}$ is an $(\ell-1)$-independent $d$-Euclidean structure over
$\Omega$ satisfying $\texttt{UCouple}[\ell]$, then for the associated
structure $\widehat{{\mathcal{N}}}$ over $\Omega^{2}$ given by (9.2), whenever
$\lvert A\rvert\leq\ell$, there exists a unique amalgamating
$(A,V,\ell)$-type. In turn, this uniqueness was obtained as a consequence of
uniqueness of overlapping $(A,V)$-types given by Proposition 5.8.

Let us now prove the deferred property (e).

###### Lemma 9.4.

Within Definition 9.2, property (e) holds, that is, for every injection
$\beta\colon A^{\prime}\to A$ between finite sets $A^{\prime}$ and $A$, both
disjoint from a countably infinite set $V$, the diagram

${{\mathcal{E}}_{A}^{(d)}(\Omega^{2})}$${{\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})}$${{\mathcal{S}}_{A,V}^{(d),\operatorname{dsct},\ell}(\Omega^{2})}$${{\mathcal{K}}_{A}}$${{\mathcal{E}}_{A^{\prime}}^{(d)}(\Omega^{2})}$${{\mathcal{S}}_{A^{\prime},V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})}$${{\mathcal{S}}_{A^{\prime},V}^{(d),\operatorname{dsct},\ell}(\Omega^{2})}$${{\mathcal{K}}_{A^{\prime}}}$$\scriptstyle{\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}}$$\scriptstyle{\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{A,V}}$$\scriptstyle{M^{\widehat{{\mathcal{N}}}}_{A}}$$\scriptstyle{\beta^{*}}$$\scriptstyle{\pi^{\operatorname{amlg},\ell}_{A,V}}$$\scriptstyle{M^{\operatorname{amlg},\ell}_{A,V}}$$\scriptstyle{\beta^{*}}$$\scriptstyle{M^{\operatorname{dsct}}_{A,V}}$$\scriptstyle{\beta^{*}}$$\scriptstyle{\beta^{*}}$$\scriptstyle{\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}}$$\scriptstyle{\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{A^{\prime},V}}$$\scriptstyle{M^{\widehat{{\mathcal{N}}}}_{A^{\prime}}}$$\scriptstyle{\pi^{\operatorname{amlg},\ell}_{A^{\prime},V}}$$\scriptstyle{M^{\operatorname{amlg},\ell}_{A^{\prime},V}}$$\scriptstyle{M^{\operatorname{dsct}}_{A^{\prime},V}}$

is weakly commutative.

###### Proof.

To completely prove the lemma, it suffices to show all weakly commutative
properties involving only the first row and to show weak equivariance of
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}$,
$\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}$,
$M^{\widehat{{\mathcal{N}}}}$, $\pi^{\operatorname{amlg},\ell}$,
$M^{\operatorname{amlg},\ell}$ and $M^{\operatorname{dsct}}$.

Equivariance of $\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}$,
$M^{\widehat{{\mathcal{N}}}}$ and $M^{\operatorname{dsct}}$ was shown in Lemma
5.7 and equivariance of $\pi^{\operatorname{amlg},\ell}$ and
$M^{\operatorname{amlg},\ell}$ was observed in (9.1). The final weak
equivariance of
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}$ will be the
last thing we prove.

Let us prove the weakly commutative properties involving the first row. The
equalities

$\displaystyle
M^{\operatorname{dsct}}_{A,V}\mathbin{\circ}\pi^{\operatorname{amlg},\ell}_{A,V}$
$\displaystyle=M^{\operatorname{amlg},\ell}_{A,V},$ $\displaystyle
M^{\operatorname{dsct}}_{A,V}\mathbin{\circ}\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{A,V}$
$\displaystyle=M^{\widehat{{\mathcal{N}}}}_{A},$

follow straightforwardly from definitions (the latter already appeared in
Lemma 5.7).

We now show

$\displaystyle\pi^{\operatorname{amlg},\ell}_{A,V}\mathbin{\circ}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$
$\displaystyle=\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{A,V}$

within the domain of the left-hand side.

When $\lvert A\rvert\leq\ell$, then by definition both sides evaluate to the
unique dissociated $(A,V)$-type $q_{A}$ within the domain of the left-hand
side.

When $\lvert A\rvert>\ell$, for
$(x,(y,z))\in\operatorname{dom}(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V})$
where $x\in{\mathcal{E}}_{A}^{(d)}(\Omega^{2})$,
$y\in\Omega\times{\mathcal{O}}_{A}^{d}$ and $z\in\Omega$, we have

$\displaystyle\pi^{\operatorname{amlg},\ell}_{A,V}\bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(x,(y,z))\bigr{)}$
$\displaystyle=\pi^{\operatorname{amlg},\ell}_{A,V}\biggl{(}(\alpha_{A}^{-1})^{*}\Bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{\lvert
A\rvert,V}\bigl{(}\alpha_{A}^{*}(x,(y,z))\bigr{)}\Bigr{)}\biggr{)}$
$\displaystyle=(\alpha_{A}^{-1})^{*}\Biggl{(}\pi^{\operatorname{amlg},\ell}_{\lvert
A\rvert,V}\biggl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{\lvert
A\rvert,V}\Bigl{(}\alpha_{A}^{*}(x),\bigl{(}\alpha_{A}^{*}(y),z\bigr{)}\Bigr{)}\biggr{)}\Biggr{)}$
$\displaystyle=(\alpha_{A}^{-1})^{*}\Biggl{(}\pi\Biggl{(}G\biggl{(}x,\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{\lvert
A\rvert,V}\Bigl{(}\alpha_{A}^{*}(x),\bigl{(}\alpha_{A}^{*}(y),z\bigr{)}\Bigr{)},z\biggr{)}\Biggr{)}\Biggr{)}$
$\displaystyle=(\alpha_{A}^{-1})^{*}\Biggl{(}g\biggl{(}x,\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{\lvert
A\rvert,V}\Bigl{(}\alpha_{A}^{*}(x),\bigl{(}\alpha_{A}^{*}(y),z\bigr{)}\Bigr{)}\biggr{)}\Biggr{)}$
$\displaystyle=(\alpha_{A}^{-1})^{*}\Biggl{(}\pi^{\operatorname{ovlp}}_{\lvert
A\rvert,V}\biggl{(}\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{\lvert
A\rvert,V}\Bigl{(}\alpha_{A}^{*}(x),\bigl{(}\alpha_{A}^{*}(y),z\bigr{)}\Bigr{)}\biggr{)}\Biggr{)}$
$\displaystyle=(\alpha_{A}^{-1})^{*}\biggl{(}\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{\lvert
A\rvert,V}\Bigl{(}\alpha_{A}^{*}(x),\bigl{(}\alpha_{A}^{*}(y),z\bigr{)}\Bigr{)}\biggr{)}$
$\displaystyle=\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{A,V}\bigl{(}x,(y,z)\bigr{)},$

where the second equality follows from the already shown equivariance of
$\pi^{\operatorname{amlg},\ell}$, in the third equality, the undecorated $\pi$
is the projection $\pi\colon V\times W\to V$ of Lemma 8.3 (i.e.,
$\pi\colon{\mathcal{S}}_{\lvert
A\rvert,V}^{(d),\operatorname{dsct}}\times[0,1]\to{\mathcal{S}}_{\lvert
A\rvert,V}^{(d),\operatorname{dsct}}$), the fourth equality follows from Lemma
8.3(v), the fifth equality follows from the definition of $g$ in (9.4), the
sixth equality follows from Lemma 5.7 and the final equality follows from the
already shown equivariance of
$\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}$ (also shown in
Lemma 5.7).

To conclude the weak commutativity of the first row, it remains to show

$\displaystyle
M^{\operatorname{amlg},\ell}_{A,V}\mathbin{\circ}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$
$\displaystyle=M^{\widehat{{\mathcal{N}}}}_{A,V}$

within the domain of the left-hand side. This follows from the other already
shown weak commutativities of the first row:

$\displaystyle
M^{\operatorname{amlg},\ell}_{A,V}\mathbin{\circ}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$
$\displaystyle=M^{\operatorname{dsct},\ell}_{A,V}\mathbin{\circ}\pi^{\operatorname{amlg},\ell}_{A,V}\mathbin{\circ}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}=M^{\operatorname{dsct},\ell}_{A,V}\mathbin{\circ}\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{A,V}=M^{\widehat{{\mathcal{N}}}}_{A,V}.$

It remains to show weak equivariance of
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}$, that is, we
need to show that if $\beta\colon A^{\prime}\to A$ is an injection, then

$\displaystyle\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}\mathbin{\circ}\beta^{*}$
$\displaystyle=\beta^{*}\mathbin{\circ}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$

within the common domain of the two sides.

Let $x\in{\mathcal{E}}_{A^{\prime}}^{(d)}(\Omega^{2})$ be a point in the
common domain of the two sides.

We first consider the case when $A^{\prime}$ and $A$ have the same
cardinality, say $a$, and we prove the weak equivariance by induction in $a$.

If $a\leq\ell$ (and assuming inductively that weak equivariance holds for
smaller sizes), we note that

$\displaystyle(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}\mathbin{\circ}\beta^{*})(x)$
$\displaystyle=\Biggl{(}\biggl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V}\Bigl{(}\iota_{B,A^{\prime}}^{*}\bigl{(}\beta^{*}(x)\bigr{)}\Bigr{)}\mathrel{\bigg{|}}B\in
r(A^{\prime},a-1)\biggr{)},q_{A^{\prime}}\Biggr{)}$
$\displaystyle=\Biggl{(}\biggl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V}\Bigl{(}\beta\mathord{\downarrow}_{B}^{*}\bigl{(}\iota_{\beta(B),A}^{*}(x)\bigr{)}\Bigr{)}\mathrel{\bigg{|}}B\in
r(A^{\prime},a-1)\biggr{)},q_{A^{\prime}}\Biggr{)}$
$\displaystyle=\Biggl{(}\biggl{(}\beta\mathord{\downarrow}_{B}^{*}\Bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{\beta(B),V}\bigl{(}\iota_{\beta(B),A}^{*}(x)\bigr{)}\Bigr{)}\mathrel{\bigg{|}}B\in
r(A^{\prime},a-1)\biggr{)},\beta^{*}(q_{A})\Biggr{)}$
$\displaystyle=\beta^{*}\biggl{(}\Bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V}\bigl{(}\iota_{B,A}^{*}(x)\bigr{)}\mathrel{\Big{|}}B\in
r(A,\lvert A\rvert-1)\Bigr{)},q_{A}\biggr{)}$
$\displaystyle=\beta^{*}\bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(x)\bigr{)},$

where the second equality follows from contra-variance and the fact that
$\beta\mathbin{\circ}\iota_{B,A^{\prime}}=\iota_{\beta(B),A}\mathbin{\circ}\beta\mathord{\downarrow}_{B}$
and the third equality follows from inductive hypothesis along with
$\beta^{*}(q_{A})=q_{A^{\prime}}$ (as $q_{A}$ and $q_{A^{\prime}}$ are the
unique up to zero measure overlapping types of $\widehat{{\mathcal{N}}}$ over
their respective sets).

Suppose now that $a>\ell$ and consider first the case $A=A^{\prime}=[a]$. Then
a similar derivation holds using $S_{a}$-equivariance of $G$ instead: writing
$x=(w,(y,z))$, where $w\in{\mathcal{E}}_{a,a-1}^{(d)}(\Omega^{2})$,
$y\in\Omega\times{\mathcal{O}}_{a}^{d}$ and $z\in\Omega$, we have

$\displaystyle\\!\\!\\!\\!\\!\\!(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}\mathbin{\circ}\beta^{*})(x)$
$\displaystyle=\Biggl{(}\biggl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V}\Bigl{(}\iota_{B,a}^{*}\bigl{(}\beta^{*}(x)\bigr{)}\Bigr{)}\mathrel{\bigg{|}}B\in
r(a,a-1)\biggr{)},G\Bigl{(}\beta^{*}(x),\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{a,V}\bigl{(}\beta^{*}(w),(\beta^{*}(y),z)\bigr{)},z\Bigr{)}\Biggr{)}$
$\displaystyle=\Biggl{(}\biggl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V}\Bigl{(}\beta\mathord{\downarrow}_{B}^{*}\bigl{(}\iota_{\beta(B),a}^{*}(x)\bigr{)}\Bigr{)}\mathrel{\bigg{|}}B\in
r(a,a-1)\biggr{)},\beta^{*}\biggl{(}G\Bigl{(}x,\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{a,V}\bigl{(}w,(y,z)\bigr{)},z\Bigr{)}\biggr{)}\Biggr{)}$
$\displaystyle=\Biggl{(}\beta^{*}\biggl{(}\Bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V}\bigl{(}\iota_{\beta(B),a}^{*}(x)\bigr{)}\Bigr{)}\biggr{)}\mathrel{\bigg{|}}B\in
r(a^{\prime},a-1)\biggr{)},\beta^{*}\biggl{(}G\Bigl{(}x,\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{a,V}\bigl{(}w,(y,z)\bigr{)},z\Bigr{)}\biggr{)}\Biggr{)}$
$\displaystyle=(\beta^{*}\mathbin{\circ}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V})(x),$

where the second equality follows from $S_{a}$-equivariance of $G$ and
$\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}$ (from Lemma 5.7)
and the fact that
$\beta\mathbin{\circ}\iota_{B,a}=\iota_{\beta(B),a}\mathbin{\circ}\beta\mathord{\downarrow}_{B}$
and the third equality follows from inductive hypothesis.

We now consider the case when $\lvert A\rvert=\lvert A^{\prime}\rvert>\ell$
(but are not necessarily of the form $[a]$). Letting
$a\stackrel{{\scriptstyle\text{def}}}{{=}}\lvert A\rvert$ and
$\gamma\stackrel{{\scriptstyle\text{def}}}{{=}}\alpha_{A}^{-1}\mathbin{\circ}\beta\mathbin{\circ}\alpha_{A^{\prime}}\in
S_{a}$, we have

$\displaystyle(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}\mathbin{\circ}\beta^{*})(x)$
$\displaystyle=\bigl{(}(\alpha_{A^{\prime}}^{-1})^{*}\mathbin{\circ}\mathbbm{p}_{a,V}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}\mathbin{\circ}\alpha_{A^{\prime}}^{*}\mathbin{\circ}\beta^{*}\bigr{)}(x)=\bigl{(}(\alpha_{A^{\prime}}^{-1})^{*}\mathbin{\circ}\mathbbm{p}_{a,V}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}\mathbin{\circ}\gamma^{*}\mathbin{\circ}\alpha_{A}^{*}\bigr{)}(x)$
$\displaystyle=\bigl{(}(\alpha_{A^{\prime}}^{-1})^{*}\mathbin{\circ}\gamma^{*}\mathbin{\circ}\mathbbm{p}_{a,V}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}\mathbin{\circ}\alpha_{A}^{*}\bigr{)}(x)=\bigl{(}\beta^{*}\mathbin{\circ}(\alpha_{A}^{-1})^{*}\mathbin{\circ}\mathbbm{p}_{a,V}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}\mathbin{\circ}\alpha_{A}^{*}\bigr{)}(x)$
$\displaystyle=(\beta^{*}\mathbin{\circ}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V})(x),$

where the third equality follows from the previous case (for $[a]$).

This concludes all cases in which $\lvert A^{\prime}\rvert=\lvert A\rvert$
(i.e., cases in which $\beta$ is a bijection).

We now prove the case when $A^{\prime}=[a^{\prime}]$ and $A=[a]$ for some
$a^{\prime},a\in\mathbb{N}$ with $a^{\prime}<a$. We need to show that for
every $C\in r(a^{\prime})$ we have

$\displaystyle(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}\mathbin{\circ}\beta^{*})(x)_{C}$
$\displaystyle=(\beta^{*}\mathbin{\circ}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V})(x)_{C}.$

If $\lvert C\rvert<a^{\prime}$, then we have

$\displaystyle(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}\mathbin{\circ}\beta^{*})(x)_{C}$
$\displaystyle=\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{C,V}\Bigl{(}\iota_{C,[a^{\prime}]}^{*}\bigl{(}\beta^{*}(x)\bigr{)}\Bigr{)}=\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{C,V}\Bigl{(}\beta\mathord{\downarrow}_{C}^{*}\bigl{(}\iota_{\beta(C),[a]}^{*}(x)\bigr{)}\Bigr{)}$
$\displaystyle=\beta\mathord{\downarrow}_{C}^{*}\Bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{\beta(C),V}\bigl{(}\iota_{\beta(C),[a]}^{*}(x)\bigr{)}\Bigr{)}=\beta\mathord{\downarrow}_{C}^{*}\bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}(x)_{\beta(C)}\bigr{)}=\beta^{*}\bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}(x)\bigr{)}_{C},$

where the third equality follows from the already proved case as
$\beta\mathord{\downarrow}_{C}$ is bijective.

If $C=[a^{\prime}]$, then we have

$\displaystyle(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}\mathbin{\circ}\beta^{*})(x)_{C}$
$\displaystyle=\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a^{\prime},V}\bigl{(}\beta^{*}(x)\bigr{)}_{[a^{\prime}]}=\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a^{\prime},V}\Bigl{(}\beta\mathord{\downarrow}_{[a^{\prime}]}^{*}\bigl{(}\iota_{\beta([a^{\prime}]),[a]}^{*}(x)\bigr{)}\Bigr{)}_{[a^{\prime}]}$
$\displaystyle=\beta\mathord{\downarrow}_{[a^{\prime}]}^{*}\Bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{\beta([a^{\prime}]),V}\bigl{(}\iota_{\beta([a^{\prime}]),[a]}^{*}(x)\bigr{)}\Bigr{)}_{[a^{\prime}]}=\beta\mathord{\downarrow}_{[a^{\prime}]}^{*}\bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}(x)_{\beta([a^{\prime}])}\bigr{)}_{[a^{\prime}]}$
$\displaystyle=\beta\mathord{\downarrow}_{[a^{\prime}]}^{*}\Bigl{(}\bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}(x)_{\beta([a^{\prime}])}\bigr{)}_{\beta([a^{\prime}])}\Bigr{)}=\beta^{*}\bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(x)\bigr{)}_{C},$

where the third equality follows from the already proved case as
$\beta\mathord{\downarrow}_{[a^{\prime}]}$ is bijective, the fifth equality
follows from the contra-variant definition of
$\beta\mathord{\downarrow}_{[a^{\prime}]}^{*}$ and the sixth equality follows
from the contra-variant definition of $\beta^{*}$.

It remains to show the case when $A^{\prime}$ and $A$ are such that $\lvert
A^{\prime}\rvert<\lvert A\rvert$ and are not necessarily of the form
$[a^{\prime}]$ and $[a]$, respectively. Let
$a^{\prime}\stackrel{{\scriptstyle\text{def}}}{{=}}\lvert A^{\prime}\rvert$
and $a\stackrel{{\scriptstyle\text{def}}}{{=}}\lvert A\rvert$, consider the
injection
$\gamma\stackrel{{\scriptstyle\text{def}}}{{=}}\alpha_{A}^{-1}\mathbin{\circ}\beta\mathbin{\circ}\alpha_{A^{\prime}}\colon[a^{\prime}]\to[a]$
and note that

$\displaystyle(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}\mathbin{\circ}\beta^{*})(x)$
$\displaystyle=\bigl{(}(\alpha_{A^{\prime}}^{-1})^{*}\mathbin{\circ}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a^{\prime},V}\mathbin{\circ}\alpha_{A^{\prime}}^{*}\mathbin{\circ}\beta^{*}\bigr{)}(x)=\bigl{(}(\alpha_{A^{\prime}}^{-1})^{*}\mathbin{\circ}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a^{\prime},V}\mathbin{\circ}\gamma^{*}\mathbin{\circ}\alpha_{A}^{*}\bigr{)}(x)$
$\displaystyle=\bigl{(}(\alpha_{A^{\prime}}^{-1})^{*}\mathbin{\circ}\gamma^{*}\mathbin{\circ}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a^{\prime},V}\mathbin{\circ}\alpha_{A}^{*}\bigr{)}(x)=\bigl{(}\beta^{*}\mathbin{\circ}(\alpha_{A}^{-1})^{*}\mathbin{\circ}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a^{\prime},V}\mathbin{\circ}\alpha_{A}^{*}\bigr{)}(x)$
$\displaystyle=\beta^{*}\bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(x)\bigr{)},$

where the third equality follows from the already proved case as
$\gamma\colon[a^{\prime}]\to[a]$.

This concludes weak equivariance of
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}$. ∎

The next lemma proves the weak amalgamation property (item (vi) of Definition
9.2), completing Definition 9.2, and proves the strong amalgamation that we
have been promising all along.

###### Lemma 9.5.

Let ${\mathcal{N}}$ be an $(\ell-1)$-independent $d$-Euclidean structure in
${\mathcal{L}}$ over $\Omega$ satisfying $\texttt{UCouple}[\ell]$, let $V$ be
a countably infinite set and let $A$ be a finite set disjoint from $V$. Let
also $\widehat{{\mathcal{N}}}$ be the $d$-Euclidean structure in
${\mathcal{L}}$ over $\Omega^{2}$ as in Definition 9.2, i.e.,
$\widehat{{\mathcal{N}}}$ is given by

$\displaystyle\widehat{{\mathcal{N}}}_{P}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\\{(x,x^{\prime})\in{\mathcal{E}}_{k(P)}^{(d)}(\Omega)\times{\mathcal{E}}_{k(P)}(\Omega)\mid
x\in{\mathcal{N}}_{P}\\}\qquad(P\in{\mathcal{L}}).$

Finally, let $k\in\\{0,\ldots,\lvert A\rvert-1\\}$.

Then the following hold.

  1. i.

Weak amalgamation: if $\bm{x}$ is sampled in
${\mathcal{E}}_{A,k}^{(d)}(\Omega^{2})$ according to $\mu^{(d)}$, then
$\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{A,V}(\bm{x})$ is
conditionally independent from $\tau_{A,k}(\bm{x})$ given

$\displaystyle\bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}(\iota_{A^{\prime},A}(\bm{x}))\mathrel{\big{|}}A^{\prime}\in
r(A,k)\bigr{)},$

where
$\tau_{A,k}\colon{\mathcal{E}}_{A}^{(d)}(\Omega^{2})\to{\mathcal{E}}_{A,k}^{(d)}(\Omega^{2})$
is the natural projection.

  2. ii.

Amalgamating types are well-defined:
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$,
$\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$ and
$\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$ are well-
defined.

  3. iii.

Strong amalgamation: if $\bm{x}$ is sampled in
${\mathcal{E}}_{A}^{(d)}(\Omega^{2})$ according to $\mu^{(d)}$, then
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(\bm{x})$
is conditionally independent from $\tau_{A,k}(\bm{x})$ given

$\displaystyle\bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}(\iota_{A^{\prime},A}(\bm{x}))\mathrel{\big{|}}A^{\prime}\in
r(A,k)\bigr{)},$

where
$\tau_{A,k}\colon{\mathcal{E}}_{A}^{(d)}(\Omega^{2})\to{\mathcal{E}}_{A,k}^{(d)}(\Omega^{2})$
is the natural projection.

###### Proof.

We will prove all assertions by induction on $\lvert A\rvert$ and the logic of
the inductive step is:

  1. A.

Item (i) for a set $A$ and $k=\lvert A\rvert-1$ implies item (ii) for $A$.

  2. B.

Item (ii) for all sets of size at most $\lvert A\rvert$ implies item (iii) for
$A$.

  3. C.

Item (iii) for all sets of size at most $\lvert A\rvert-1$ implies item (i)
for $A$ (and every $k<\lvert A\rvert$).

Item (A) is trivial as item (i) when $k=\lvert A\rvert-1$ is exactly item (vi)
of Definition 9.2.

For item (B), we consider three random variables $\bm{p}$, $\bm{q}$ and
$\bm{r}$ in $\prod_{A^{\prime}\in
r(A)}{\mathcal{S}}_{A^{\prime},V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$.

For $\bm{p}$, we simply let

$\displaystyle\bm{p}_{A^{\prime}}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}\bigl{(}\iota_{A^{\prime},A}^{*}(\bm{x})\bigr{)}\qquad(A^{\prime}\in
r(A)).$

For $\bm{q}$, we inductively define

$\displaystyle\bm{q}_{A^{\prime}}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}\bigl{(}\bm{q}_{C}\mid
C\in r(A^{\prime},\lvert A^{\prime}\rvert-1)\bigr{)}\qquad(A^{\prime}\in
r(A)).$

An inductive application of items (d) and (e) of Definition 9.2 shows that
$\bm{p}$ and $\bm{q}$ are equidistributed.

Finally, for $\bm{r}$, we inductively define

$\displaystyle\bm{r}_{A^{\prime}}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\begin{dcases*}\bm{p}_{A^{\prime}},&if
$\lvert A^{\prime}\rvert\leq k$,\\\
\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}\bigl{(}(\bm{r}_{C}\mid
C\in r(A^{\prime},\lvert A^{\prime}\rvert-1)\bigr{)},&if $\lvert
A^{\prime}\rvert>k$,\end{dcases*}\qquad(A^{\prime}\in r(A)).$

Since $\bm{p}$ and $\bm{q}$ are equidistributed, it follows that $\bm{r}$ also
has the same distribution as $\bm{p}$ and $\bm{q}$.

Since by construction we have $(\bm{r}_{A^{\prime}}\mid A^{\prime}\in
r(A,k))=(\bm{p}_{A^{\prime}}\mid A^{\prime}\in r(A,k))$, it follows that the
$\bm{p}$ and $\bm{r}$ have the same conditional distribution given
$(\bm{p}_{A^{\prime}}\mid A^{\prime}\in r(A,k))$. On the other hand, by
construction of $\bm{r}$, it is obvious that the conditional distribution of
$\bm{r}$ given $(\bm{p}_{A^{\prime}}\mid A^{\prime}\in r(A,k))$ is
(conditionally) independent from $(\bm{x}_{A^{\prime}}\mid A^{\prime}\in
r(A,k))$, concluding the proof of item (B).

It remains to prove item (C). We prove item (i) by induction in $k<\lvert
A\rvert$.

Let $\bm{z}$ be picked at random in $\prod_{C\in\binom{A}{>k}}(X\times
X\times{\mathcal{O}}_{C}^{d})$ according to
$\bigotimes_{C\in\binom{A}{>k}}(\mu\otimes\mu\otimes\bigotimes_{i=1}^{d}\nu_{C})$
and for each $y\in{\mathcal{E}}_{A,k}^{(d)}(\Omega^{2})$, let

$\displaystyle\bm{p}^{y}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{A,V}(y,\bm{z})$

and note that it suffices to show that there exists a $\mu^{(d)}$-measure $1$
subset $G$ of ${\mathcal{E}}_{A,k}^{(d)}(\Omega^{2})$ such that if
$y,y^{\prime}\in G$ are such that
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}(\iota_{A^{\prime},A}(y))=\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}(\iota_{A^{\prime},A}(y^{\prime}))$
for every $A^{\prime}\in r(A,k)$, then $\bm{p}^{y}$ and $\bm{p}^{y^{\prime}}$
are equidistributed. By our inductive hypothesis in $k$, we may further
suppose that $y_{A^{\prime}}=y^{\prime}_{A^{\prime}}$ for every $A^{\prime}\in
r(A,k-1)$.

We now introduce new random variables $\bm{r}^{y}$ in
${\mathcal{S}}_{A,V}^{(d),\operatorname{dsct}}(\Omega^{2})$ for each
$y\in{\mathcal{E}}_{A,k}^{(d)}(\Omega^{2})$: by Proposition 7.4, there exists
a Markov kernel

$\displaystyle\rho\colon\prod_{A^{\prime}\in\binom{A}{k}}{\mathcal{S}}_{A^{\prime},V}^{(d),\operatorname{ovlp}}(\Omega^{2})\to{\mathcal{P}}\bigl{(}{\mathcal{S}}_{A,V}^{(d),\operatorname{dsct}}(\Omega^{2})\bigr{)}$

that allows us to correctly sample the conditional distribution of the
dissociated $(A,V)$-type in $\widehat{{\mathcal{N}}}$ given the overlapping
$(A^{\prime},V)$-types for every $A^{\prime}\in\binom{A}{k}$. For each
$A^{\prime}\in\binom{A}{k}$, we let
$\tau_{A^{\prime}}\colon{\mathcal{E}}_{A^{\prime}}^{(d)}(\Omega^{2})\to{\mathcal{E}}_{A^{\prime},k-1}^{(d)}(\Omega^{2})$
be the natural projection and

$\displaystyle q^{y}_{A^{\prime}}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}\Bigl{(}\tau_{A^{\prime}}(y),\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}\bigr{(}\iota_{A^{\prime},A}^{*}(y)\bigr{)}\Bigr{)}$

be the overlapping $(A^{\prime},V)$-type recovered via
$\mathbbm{q}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}$
from the amalgamating $(A^{\prime},V,\ell)$-type of
$\iota_{A^{\prime},A}^{*}(y)$ along with the pointwise information
$\tau_{A^{\prime}}(y)$ of coordinates indexed by sets in $r(A^{\prime},k-1)$
and finally let $\bm{r}^{y}$ be sampled according to
$\rho((q^{y}_{A^{\prime}}\mid A^{\prime}\in\binom{A}{k}))$.

By item (c) in Definition 9.2, we know that

$\displaystyle q^{y}_{A^{\prime}}$
$\displaystyle=\mathbbm{p}^{\operatorname{ovlp},\widehat{{\mathcal{N}}}}_{A^{\prime},V}(\iota_{A^{\prime},A}^{*}(y))$

whenever
$\iota_{A^{\prime},A}^{*}(y)\in\operatorname{dom}(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V})$.
In turn, since our $\rho$ was provided by Proposition 7.4, we conclude that

$\displaystyle\bm{r}^{y}$
$\displaystyle\sim\mathbbm{p}^{\operatorname{dsct},\widehat{{\mathcal{N}}}}_{A,V}(y,\bm{z})=\bm{p}^{y}$
(9.6)

for $\mu^{(d)}$-almost every $y$ in the set

$\displaystyle G$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\left\\{y\in{\mathcal{E}}_{A,k}^{(d)}(\Omega^{2})\;\middle|\;\forall
A^{\prime}\in\binom{A}{k},\iota_{A^{\prime},A}^{*}(y)\in\operatorname{dom}(\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V})\right\\}.$

Clearly $\mu^{(d)}(G)=1$, so if $\widetilde{G}$ is the set of all $y\in G$
such that (9.6) holds, then $\mu^{(d)}(\widetilde{G})=1$.

Now let $y,y^{\prime}\in\widetilde{G}$ be such that
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}(\iota_{A^{\prime},A}(y))=\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}(\iota_{A^{\prime},A}(y^{\prime}))$
for every $A^{\prime}\in r(A,k)$ and $y_{C}=y^{\prime}_{C}$ for every $C\in
r(A,k-1)$. The latter condition implies that for every
$A^{\prime}\in\binom{A}{k}$, we get
$\tau_{A^{\prime}}(y)=\tau_{A^{\prime}}(y^{\prime})$, so we get
$q^{y}_{A^{\prime}}=q^{y^{\prime}}_{A^{\prime}}$, which in turn implies

$\displaystyle\bm{p}^{y}$
$\displaystyle\sim\bm{r}^{y}=\bm{r}^{y^{\prime}}\sim\bm{p}^{y^{\prime}},$

as desired. ∎

Before we proceed, let us show equivariance of the definition of
$\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}$ with respect to
bijections.

###### Lemma 9.6.

Within Definition 9.2, if $\beta\colon A^{\prime}\to A$ is a bijection between
finite sets $A^{\prime}$ and $A$, both disjoint from a countably infinite set
$V$, then the diagram

${\prod_{B\in r(A,\lvert
A\rvert-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})}$${{\mathcal{P}}({\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2}))}$${\prod_{B\in
r(A^{\prime},\lvert
A^{\prime}\rvert-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})}$${{\mathcal{P}}({\mathcal{S}}_{A^{\prime},V}^{(d),\operatorname{amlg},\ell}(\Omega^{2}))}$$\scriptstyle{\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}}$$\scriptstyle{\beta^{*}}$$\scriptstyle{\beta^{*}}$$\scriptstyle{\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}}$

is weakly commutative.

###### Proof.

Let $a\stackrel{{\scriptstyle\text{def}}}{{=}}\lvert A\rvert=\lvert
A^{\prime}\rvert$ and
$\gamma\stackrel{{\scriptstyle\text{def}}}{{=}}\alpha_{A}^{-1}\mathbin{\circ}\beta\mathbin{\circ}\alpha_{A^{\prime}}\in
S_{a}$ and note that

$\displaystyle\beta^{*}\mathbin{\circ}\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$
$\displaystyle=\beta^{*}\mathbin{\circ}(\alpha_{A}^{-1})^{*}\mathbin{\circ}\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}\mathbin{\circ}\alpha_{A}^{*}=(\alpha_{A^{\prime}}^{-1})^{*}\mathbin{\circ}\gamma^{*}\mathbin{\circ}\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}\mathbin{\circ}\alpha_{A}^{*}$
$\displaystyle=(\alpha_{A^{\prime}}^{-1})^{*}\mathbin{\circ}\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}\mathbin{\circ}\gamma^{*}\mathbin{\circ}\alpha_{A}^{*}=(\alpha_{A^{\prime}}^{-1})^{*}\mathbin{\circ}\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}\mathbin{\circ}\alpha_{A^{\prime}}^{*}\mathbin{\circ}\beta^{*}$
$\displaystyle=\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A^{\prime},V}\mathbin{\circ}\beta^{*},$

where the third equality follows since
$\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}$ is
$S_{a}$-equivariant. ∎

We can finally prove the implication (i)$\implies$(iii) of Theorem 2.2.

###### Proposition 9.7 (Theorem 2.2(i)$\implies$(iii)).

If $\phi\in\operatorname{Hom}^{+}({\mathcal{A}}[T],\mathbb{R})$ satisfies
$\texttt{UCouple}[\ell]$, then there exists an $\ell$-independent
$1$-Euclidean structure ${\mathcal{N}}$ such that $\phi=\phi_{\mathcal{N}}$.

###### Proof.

Let $\widetilde{\Omega}=([0,1],\lambda)$, where $\lambda$ is the Lebesgue
measure. Since $\texttt{UCouple}[\ell]$ implies $\texttt{UCouple}[\ell-1]$ and
since every $d$-Euclidean structure is $0$-independent, it suffices to show
that if ${\mathcal{N}}$ is an $(\ell-1)$-independent $d$-Euclidean structure
over some $\Omega=(X,\lambda)$, then there exists an $\ell$-independent
$1$-Euclidean structure ${\mathcal{H}}$ over $\widetilde{\Omega}$ such that
$\phi_{\mathcal{N}}=\phi_{\mathcal{H}}$ (in fact, it would even suffice to
show only the case $d=1$ and $\Omega=\widetilde{\Omega}$, but the general
$d\in\mathbb{N}$ and arbitrary $\Omega$ case is not any harder and the
different notation will help keep track of which points are in which spaces).

Let $\widehat{{\mathcal{N}}}$ be the $d$-Euclidean structure in
${\mathcal{L}}$ over $\Omega^{2}$ as in Definition 9.2:

$\displaystyle\widehat{{\mathcal{N}}}_{P}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\\{(x,x^{\prime})\in{\mathcal{E}}_{k(P)}^{(d)}(\Omega)\times{\mathcal{E}}_{k(P)}(\Omega)\mid
x\in{\mathcal{N}}_{P}\\}\qquad(P\in{\mathcal{L}}).$

We will first provide an alternative way of sampling amalgamating
$(A,V,\ell)$-types of $\widehat{{\mathcal{N}}}$ for every finite
$A\subseteq\mathbb{N}_{+}$ using i.i.d. variables distributed uniformly in
$[0,1]$ (note that by equivariance of Lemma 9.4, this also provides a way of
sampling the amalgamating $(A,V,\ell)$-types even when $A$ is not a subset of
$\mathbb{N}_{+}$). Throughout this proof, we always assume $V$ is a countably
infinite set disjoint from $\mathbb{N}_{+}$ (which ensures it is disjoint from
any $A$ we consider) and we also follow the notation of Definition 9.2.

First, since all spaces of amalgamating types are Borel spaces (see
Proposition 6.6(ii)), for each $a\in\mathbb{N}_{+}$, we can associate to each
Markov kernel

$\displaystyle\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}\colon\prod_{B\in
r(a,a-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$
$\displaystyle\rightharpoonup{\mathcal{P}}({\mathcal{S}}_{a,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2}))$
of Definition 9.2 a measurable partial function $\displaystyle
F_{a,V}\colon\prod_{B\in
r(a,a-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})\times[0,1]$
$\displaystyle\rightharpoonup{\mathcal{S}}_{a,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$

such that

$\displaystyle F_{a,V}(p,\mathord{-})_{*}(\lambda)$
$\displaystyle=\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}(p)$
(9.7)

for every $p\in\prod_{B\in
r(a,a-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$, that
is, $\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}(p)$ is
the distribution of $F_{a,V}(p,\bm{t})$ where $\bm{t}$ is sampled uniformly at
random in $[0,1]$.

Item (d) of Definition 9.2 says that when $a\leq\ell$, then
$\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}$ is the
constant function that maps all points to the distribution concentrated on the
unique point $q_{a}$ in the image of
$\mathbbm{p}^{(d),\operatorname{amlg},\ell}_{a,V}$, so when $a\leq\ell$, we
can further require that $F_{a,V}$ is the constant function that maps every
point to $q_{a}$.

It is worth noting that even though
$\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{a,V}$ is
$S_{a}$-equivariant, we cannot require $F_{a,V}$ to be $S_{a}$-equivariant (as
there are $S_{a}$-invariant distributions that cannot be generated by
$S_{a}$-equivariant functions; e.g., the quasirandom tournament, see Example
1.4).

For $A\subseteq\mathbb{N}_{+}$ finite non-empty that is not of the form $[a]$
for some $a\in\mathbb{N}_{+}$, we also define

$\displaystyle F_{A,V}\colon\prod_{B\in r(A,\lvert
A\rvert-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})\times[0,1]$
$\displaystyle\rightharpoonup{\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$

in a similar fashion to (9.5):

$\displaystyle F_{A,V}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(\alpha_{A}^{-1})^{*}\mathbin{\circ}F_{\lvert
A\rvert,V}\mathbin{\circ}\alpha_{A}^{*},$ (9.8)

where $\alpha_{A}\colon[\lvert A\rvert]\to A$ is the same bijection used in
(9.5) in Definition 9.2 and we declare the rightmost $\alpha_{A}^{*}$ in the
above to act trivially on the coordinate in $[0,1]$. Note also that since we
set $\alpha_{[a]}=\operatorname{id}_{a}$, it follows that (9.8) for every
$A\subseteq\mathbb{N}_{+}$ regardless of whether it is of the form $[a]$ or
not.

Note that Lemma 9.6 along with (9.7) implies that

$\displaystyle F_{A,V}(p,\mathord{-})_{*}(\lambda)$
$\displaystyle=\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(p)$
(9.9)

for every $p\in\prod_{B\in r(A,\lvert
A\rvert-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$.

Note also that equivariance of
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}$ (Lemma 9.4)
implies that $F_{A,V}$ also has the property that if $\lvert A\rvert\leq\ell$,
then $F_{A,V}$ is the constant function that maps every point to the unique
point in the image of
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$.

Again, $F_{A,V}$ is not necessarily $S_{A}$-equivariant. To address this, we
will use order variables to define a partial function

$\displaystyle G_{A,V}\colon\prod_{B\in r(A,\lvert
A\rvert-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})\times[0,1]\times{\mathcal{O}}_{A}$
$\displaystyle\rightharpoonup{\mathcal{S}}_{A,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$

by

$\displaystyle G_{A,V}(p,t,{\lhd})$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}(\sigma_{\lhd}^{-1})^{*}\Bigl{(}F_{A,V}\bigl{(}\sigma_{\lhd}^{*}(p,t)\bigr{)}\Bigr{)}\qquad\left(p\in\prod_{B\in
r(A,\lvert
A\rvert-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2}),t\in[0,1],{\lhd}\in{\mathcal{O}}_{A}\right),$

where $\sigma_{\lhd}\in S_{A}$ is the unique permutation such that

$\displaystyle(\sigma_{\lhd}\mathbin{\circ}\alpha_{A})^{*}({\lhd})$
$\displaystyle={\leq_{\lvert A\rvert}},$

where $\leq_{\lvert A\rvert}$ is the usual order on $[\lvert A\rvert]$
(recalling that $\alpha_{A}\colon[\lvert A\rvert]\to A$ is the bijection used
in (9.5) in Definition 9.2). Again, Lemma 9.4 implies that $G_{A,V}$ also has
the property that if $\lvert A\rvert\leq\ell$, then $G_{A,V}$ is the constant
function that maps every point to the unique point in the image of
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}$.

The next claim gives a form of equivariance of $\sigma_{\lhd}$.

###### Claim 9.7.1.

If $\beta\colon A\to B$ is a bijection and ${\lhd}\in{\mathcal{O}}_{B}$, then

$\displaystyle\sigma_{\beta^{*}({\lhd})}$
$\displaystyle=\beta^{-1}\mathbin{\circ}\sigma_{\lhd}\mathbin{\circ}\alpha_{B}\mathbin{\circ}\alpha_{A}^{-1}.$

###### Proof of Claim 9.7.1..

Letting $a\stackrel{{\scriptstyle\text{def}}}{{=}}\lvert A\rvert=\lvert
B\rvert$, the claim follows from:

$\displaystyle(\beta^{-1}\mathbin{\circ}\sigma_{\lhd}\mathbin{\circ}\alpha_{B}\mathbin{\circ}\alpha_{A}^{-1}\mathbin{\circ}\alpha_{A})^{*}\bigl{(}\beta^{*}({\lhd})\bigr{)}$
$\displaystyle=(\sigma_{\lhd}\mathbin{\circ}\alpha_{B})^{*}({\lhd})={\leq_{a}}.\qed$

The next claim says that the partial functions $G_{A,V}$ are equivariant under
bijections.

###### Claim 9.7.2.

If $\beta\colon A\to B$ is a bijection, then the diagram

${\prod_{C\in r(B,\lvert
B\rvert-1)}{\mathcal{S}}_{C,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})\times{\mathcal{O}}_{B}}$${{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})}$${\prod_{C\in
r(B,\lvert
B\rvert-1)}{\mathcal{S}}_{C,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})\times{\mathcal{O}}_{B}}$${{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})}$$\scriptstyle{G_{B,V}}$$\scriptstyle{\beta^{*}}$$\scriptstyle{\beta^{*}}$$\scriptstyle{G_{B,V}}$

is weakly commutative.

###### Proof of Claim 9.7.2..

Let $a\stackrel{{\scriptstyle\text{def}}}{{=}}\lvert A\rvert=\lvert B\rvert$
and let
$(p,t,{\lhd})\in\operatorname{dom}(G_{A,V}\mathbin{\circ}\beta^{*})\cap\operatorname{dom}(G_{B,V})$
and note that

$\displaystyle(G_{A,V}\mathbin{\circ}\beta^{*})(p,t,{\lhd})$
$\displaystyle=(\sigma^{-1}_{\beta^{*}({\lhd})})^{*}\biggl{(}F_{A,V}\Bigl{(}\sigma_{\beta^{*}({\lhd})}\bigl{(}\beta^{*}(p,t)\bigr{)}\Bigr{)}\biggr{)}$
$\displaystyle=(\alpha_{A}\mathbin{\circ}\alpha_{B}^{-1}\mathbin{\circ}\sigma_{\lhd}^{-1}\mathbin{\circ}\beta)^{*}\biggl{(}F_{A,V}\Bigl{(}(\beta^{-1}\mathbin{\circ}\sigma_{\lhd}\mathbin{\circ}\alpha_{B}\mathbin{\circ}\alpha_{A}^{-1})^{*}\bigl{(}\beta^{*}(p,t)\bigr{)}\Bigr{)}\biggr{)}$
$\displaystyle=\beta^{*}\biggl{(}(\alpha_{A}\mathbin{\circ}\alpha_{B}^{-1}\mathbin{\circ}\sigma_{\lhd}^{-1})^{*}\Bigl{(}F_{A,V}\bigl{(}(\sigma_{\lhd}\mathbin{\circ}\alpha_{B}\mathbin{\circ}\alpha_{A}^{-1})^{*}(p,t)\bigr{)}\Bigr{)}\biggr{)}$
$\displaystyle=\beta^{*}\biggl{(}(\alpha_{B}^{-1}\mathbin{\circ}\sigma_{\lhd}^{-1})^{*}\Bigl{(}F_{a,V}\bigl{(}(\sigma_{\lhd}\mathbin{\circ}\alpha_{B})^{*}(p,t)\bigr{)}\Bigr{)}\biggr{)}$
$\displaystyle=\beta^{*}\biggl{(}(\sigma_{\lhd}^{-1})^{*}\Bigl{(}F_{B,V}\bigl{(}\sigma_{\lhd}^{*}(p,t)\bigr{)}\Bigr{)}\biggr{)}$
$\displaystyle=(\beta^{*}\mathbin{\circ}G_{B,V})(p,t,{\lhd}),$

where the second equality follows from Claim 9.7.1 and the fourth and fifth
equalities follow from (9.8). ∎

We proceed to show that the partial function $G_{A,V}$ can be used to
correctly sample the conditional distribution of the amalgamating
$(A,V,\ell)$-types given the lower amalgamating types.

###### Claim 9.7.3.

For $\mu^{(d)}$-almost every $x\in{\mathcal{E}}_{A,\lvert
A\rvert-1}^{(d)}(\Omega^{2})$ and every ${\lhd}\in{\mathcal{O}}_{A}$, we have

$\displaystyle G_{A,V}(p_{x},\mathord{-},{\lhd})_{*}(\lambda)$
$\displaystyle=\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(p_{x}),$

where $p_{x}\in\prod_{B\in r(A,\lvert
A\rvert-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$ is
given by

$\displaystyle(p_{x})_{B}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}\bigl{(}\iota_{B,A}^{*}(x)\bigr{)}\qquad(B\in
r(A,\lvert A\rvert-1)).$

In particular, we have

$\displaystyle G_{A,V}(p_{x},\mathord{-})_{*}(\lambda\otimes\nu_{A})$
$\displaystyle=\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(p_{x})$

for $\mu^{(d)}$-almost every $x\in{\mathcal{E}}_{A,\lvert
A\rvert-1}^{(d)}(\Omega^{2})$.

###### Proof of Claim 9.7.3..

It is clear that the second assertion follows from the first as $\nu_{A}$ is
the uniform measure on ${\mathcal{O}}_{A}$.

In turn, to prove the first assertion, it suffices to show that if $\bm{x}$ is
sampled in ${\mathcal{E}}_{A,\lvert A\rvert-1}^{(d)}(\Omega^{2})$ according to
$\mu^{(d)}$, then

$\displaystyle G_{A,V}(p_{\bm{x}},\mathord{-},{\lhd})_{*}(\lambda)$
$\displaystyle=\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(p_{\bm{x}})$

with probability $1$.

By item (d) in Definition 9.2, we know that if we sample $\bm{y}$ in
$X^{2}\times{\mathcal{O}}_{A}^{d}$ according to
$\mu^{2}\otimes\bigotimes_{i=1}^{d}\nu_{A}$ independently from $\bm{x}$, then
$\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(p_{\bm{x}})$
is the conditional distribution of
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(\bm{x},\bm{y})$
given $\bm{x}$.

By Lemma 9.4, we also have

$\displaystyle\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(\bm{x},\bm{y})$
$\displaystyle=(\sigma_{\lhd}^{-1})^{*}\Bigl{(}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}\bigl{(}\sigma_{\lhd}^{*}(\bm{x},\bm{y})\bigr{)}\Bigr{)}.$

By considering conditional distributions given $\bm{x}$ of the above and
applying item (d) in Definition 9.2 again, we get

$\displaystyle\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(p_{\bm{x}})$
$\displaystyle=(\sigma_{\lhd}^{-1})^{*}\bigl{(}\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(p_{\sigma_{\lhd}^{*}(\bm{x})})\bigr{)}.$

In turn, applying (9.9) to the right-hand side of the above, we get

$\displaystyle\Theta^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{A,V}(p_{\bm{x}})$
$\displaystyle=(\sigma_{\lhd}^{-1})^{*}\Bigl{(}F_{A,V}(p_{\sigma_{\lhd}^{*}(\bm{x})},\mathord{-})_{*}(\lambda)\Bigr{)}$
$\displaystyle=((\sigma_{\lhd}^{-1})^{*}\mathbin{\circ}F_{A,V}\mathbin{\circ}\sigma_{\lhd}^{*})(p_{\bm{x}},\mathord{-})_{*}(\lambda)$
$\displaystyle=G_{A,V}(p_{\bm{x}},\mathord{-},{\lhd})_{*}(\lambda),$

where the second equality follows from equivariance of
$\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}$ (see Lemma
9.4) and since the action on $[0,1]$ is trivial. ∎

Our goal is now to upgrade the correctness of the sampling via $G_{A,V}$ to
joint distributions. To do so, for $A\in r(\mathbb{N}_{+})$ and $c\leq\lvert
A\rvert$, let us define the natural “product” partial function

$\displaystyle\widehat{G}_{A,c,V}\colon\prod_{B\in
r(A,c-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})\times\prod_{C\in\binom{A}{c}}([0,1]\times{\mathcal{O}}_{C})$
$\displaystyle\rightharpoonup\prod_{C\in\binom{A}{c}}{\mathcal{S}}_{C,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$

by

$\displaystyle\widehat{G}_{A,c,V}(p,u)_{C}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}G_{C,V}\bigl{(}\iota_{C,A}^{*}(p),u_{C}\bigr{)}$

for every $C\in\binom{A}{c}$, every $p\in\prod_{B\in
r(A,c-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$ and
every
$u\in\prod_{C^{\prime}\in\binom{A}{c}}([0,1]\times{\mathcal{O}}_{C^{\prime}})$.

The next claim is the desired upgrade of Claim 9.7.3 for
$\widehat{G}_{A,c,V}$.

###### Claim 9.7.4.

Let $A\in r(\mathbb{N}_{+})$ and let $c\leq\lvert A\rvert$. For every
$x\in{\mathcal{E}}_{A,c-1}^{(d)}(\Omega^{2})$ and every $C\in\binom{A}{c}$,
let $p_{x}\in\prod_{B\in
r(A,c-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$ and
$p_{x,C}\in\prod_{B\in
r(C,c-1)}{\mathcal{S}}_{B,V}^{(d),\operatorname{amlg},\ell}(\Omega^{2})$ be
given by

$\displaystyle(p_{x})_{B}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbbm{p}^{\operatorname{amlg},\ell,\widehat{{\mathcal{N}}}}_{B,V}\bigl{(}\iota_{B,A}^{*}(x)\bigr{)}\qquad(B\in
r(A,c-1)),$ $\displaystyle(p_{x,C})_{B}$