The following document is a translation (from French to English) of:

Gabriele H. Greco

Sur la mesurabilité d’une fonction numérique par rapport à une famille
d’ensembles

Rendiconti del Seminario Matematico della Università di Padova, tome 65
(1981), pp. 163–176.

Translated by: Jonathan M. Keith, School of Mathematics, Monash University,
<EMAIL_ADDRESS>

With thanks to: Prof. Andrea D’Agnolo, Editor-in-Chief of the above journal,
for permission to publish this translation.

# On the measurability of a numerical function with respect to a family of
sets

Gabriele H. Greco Author’s address: Department of Mathematics, University of
Trento, 38050 Povo (Trento).

## Introduction.

In this introduction and in the sequel, we will denote by the symbols $X$,
${\mathcal{P}}(X)$, $[0,+\infty]^{X}$, $\overline{{\mathbb{R}}}^{X}$,
${\mathbb{N}}$ respectively, a non-empty set, the class of subsets of $X$, the
class of functions defined on $X$ with values in $[0,+\infty]$, the class of
functions defined on $X$ with values in the extended real line
$\overline{{\mathbb{R}}}$, and the set of integers $n>0$. Following the
notations established by P. A. Meyer in [5], we will call a paving on $X$ any
family of sets ${\mathcal{E}}\subset{\mathcal{P}}(X)$ that contains the empty
set; we will call a paved set the pair $(X,{\mathcal{E}})$ comprised of a set
$X$ and a paving ${\mathcal{E}}$ on $X$; the phrase “${\mathcal{E}}$ is stable
for $(\cup f,\cap d)$” (and likewise analogous sentences) signifies that
${\mathcal{E}}$ is stable for finite unions and countable intersections. We
will also use the letters of the Greek alphabet $\alpha$, $\beta$, $\gamma$,
$\delta$ to designate monotonic set functions, that is, functions defined on
${\mathcal{P}}(X)$ with values in $[0,+\infty]$ such that
$\alpha(\emptyset)=0$ and $\alpha(A)\leqslant\alpha(B)$ if $A\subset B$. By
$\int_{X}fd\alpha$ we will designate the real number, finite or not, called
the integral of $f$ over $X$ with respect to the monotone set function
$\alpha$, defined by $\int_{X}fd\alpha=\int_{0}^{+\infty}\alpha\\{f>t\\}dt$ if
$f\in[0,+\infty]^{X}$, and by
$\int_{X}fd\alpha=\int_{X}f^{+}d\alpha-\int_{X}f^{-}d\alpha$ if
$f\in\overline{{\mathbb{R}}}^{X}$ and if $\int_{X}f^{+}d\alpha<\infty$ or
$\int_{X}f^{-}d\alpha<\infty$; if $A\in{\mathcal{P}}(X)$, we set
$\int_{A}fd\alpha=\int_{X}f\varphi_{A}d\alpha$, where $\varphi_{A}$ is the
characteristic function of the set $A$. These integrals have been studied
independently in [1] and in [3]; for their properties we will refer to [3]. If
${\mathcal{F}}$ is a filter of ${\mathcal{P}}(X)$ we will denote by
$\alpha_{{\mathcal{F}}}$ and
$\beta_{{\mathcal{F}}}:{\mathcal{P}}(X)\rightarrow\\{0,1\\}$ the set functions
defined by “$\alpha_{{\mathcal{F}}}(A)=1$ if and only if $A\in{\mathcal{F}}$”
and “$\beta_{{\mathcal{F}}}(A)=1$ if and only if $A\cap F\neq\emptyset$ for
any $F\in{\mathcal{F}}$”; for any filter ${\mathcal{F}}$ of ${\mathcal{P}}(X)$
and for any $f\in[0,+\infty]^{X}$ it holds that
$\int_{X}fd\alpha_{{\mathcal{F}}}=\liminf_{{\mathcal{F}}}f$111Translator’s
note: The original uses ‘minlim’ for ‘liminf’ and ‘maxlim’ for ‘limsup’ here
and throughout. and $\int_{X}fd\beta_{{\mathcal{F}}}=\limsup_{{\mathcal{F}}}f$
(see example $3^{\circ}$ in [3]). We set $0\cdot\infty=\infty\cdot 0=0$ and
$\inf\emptyset=+\infty$. If $(X,{\mathcal{E}})$ is a paved set, we will denote
by ${\mathcal{S}}^{+}(X,{\mathcal{E}})$ the class of functions
$\sum_{i=1}^{\infty}a_{i}\varphi_{H_{i}}$, where $a_{i}\in[0,+\infty]$,
$\\{H_{i}\\}_{i\in{\mathbb{N}}}\subset{\mathcal{E}}$ and $H_{i}\supset
H_{i+1}$ for all $i\in{\mathbb{N}}$.

In this article we propose and study a definition of measurability of a
numerical function with respect to a family of sets
${\mathcal{E}}\subset{\mathcal{P}}(X)$ that meets this requirement: “determine
numerical functions $f$ such that the integral $\int_{X}fd\alpha$ depends only
on those values of $\alpha$ on ${\mathcal{E}}$, regardless of the monotone set
function $\alpha:{\mathcal{P}}(X)\rightarrow[0,+\infty]$”; that is, we say
that a function $f\in[0,+\infty]^{X}$ is ${\mathcal{E}}$-measurable if one
always has $\int_{X}fd\alpha=\int_{X}fd\beta$, whenever the monotone set
functions $\alpha,\beta$ are such that $\alpha(H)=\beta(H)$ for all
$H\in{\mathcal{E}}$. The function $f$ is then ${\mathcal{E}}$-measurable if
and only if for each pair $a,b\in(0,+\infty)$ with $a>b$, there exists a set
$H\in{\mathcal{E}}$ such that $\\{f\geqslant a\\}\subset H\subset\\{f>b\\}$
(see Th. 1). This definition permits each paved set $(X,{\mathcal{E}})$ to be
associated with the class ${\mathcal{M}}(X,{\mathcal{E}})$ of
${\mathcal{E}}$-measurable functions, which can be: the well known class of
measurable functions with respect to a $\sigma$-algebra (if ${\mathcal{E}}$ is
a $\sigma$-algebra), the class of functions without oscillatory
discontinuities (if $X={\mathbb{R}}$ and ${\mathcal{E}}$ is the set algebra
generated by the intervals of ${\mathbb{R}}$; see Remark 2), the class of
continuous functions (if $X$ is a compact and totally disconnected Hausdorff
topological space and if ${\mathcal{E}}$ is the algebra of clopen sets), the
class of functions which are continuous almost everywhere (if $X=[0,1]$ and
${\mathcal{E}}$ is the algebra of Peano-Jordan measurable sets), the class of
$\alpha$-measurable sets in the sense of Carathéodory, with respect to a set
function $\alpha$ (if ${\mathcal{E}}$ is the algebra of $\alpha$-measurable
sets in the sense of Carathéodory; see Remark 5); in the case where
${\mathcal{E}}$ is an algebra, it follows that a function
$f\in\overline{{\mathbb{R}}}^{X}$ is ${\mathcal{E}}$-measurable if and only if
$f$ extends by continuity to the Stone space associated with the algebra
${\mathcal{E}}$.

In this article, we also notice analogies between pseudo-compact topological
spaces and semi-compact paved sets (see s. 2), and between normal topological
spaces and sets on which there are two good pavings (see s. 3).

##  1. ${\mathcal{E}}$-measurable functions.

Let $(X,{\mathcal{E}})$ be a paved set. We say that a function
$f\in[0,+\infty]^{X}$ is ${\mathcal{E}}$-measurable if we always have
$\int_{X}d\alpha=\int_{X}d\beta$, in the case where $\alpha$ and $\beta$ are
monotonic set function such that $\alpha(H)=\beta(H)$ for all
$H\in{\mathcal{E}}$. We also say that $f\in\overline{{\mathbb{R}}}^{X}$ is
${\mathcal{E}}$-measurable if $f^{+}$ and $f^{-}$ are
${\mathcal{E}}$-measurable.

We will denote by ${\mathcal{M}}(X,{\mathcal{E}})$ the class of functions
$f\in{\mathbb{R}}^{X}$ which are ${\mathcal{E}}$-measurable and we set
${\mathcal{M}}^{+}(X,{\mathcal{E}})={\mathcal{M}}(X,{\mathcal{E}})\cap[0,+\infty]^{X}$.

The simplest ${\mathcal{E}}$-measurable functions are the functions that
belong to ${\mathcal{S}}^{+}(X,{\mathcal{E}})$; indeed if
$\\{a_{i}\\}_{i\in{\mathbb{N}}}\subset[0.+\infty]$;
$\\{H_{i}\\}_{i\in{\mathbb{N}}}\subset{\mathcal{E}}$ and $H_{i+1}\subset
H_{i}$ for all $i\in{\mathbb{N}}$, we have
$\int_{X}\left(\sum_{i=1}^{\infty}a_{i}\varphi_{H_{i}}\right)d\alpha=\sum_{i=1}^{\infty}a_{i}\alpha(H_{i})$
for any monotonic set function $\alpha$.

Lemma 1. Let $(X,{\mathcal{E}})$ be a paved set and let $f\in[0,+\infty]^{X}$.
The following conditions are equivalent:

  1. i)

$f$ is ${\mathcal{E}}$-measurable,

  2. ii)

for each pair $\alpha,\beta:{\mathcal{P}}(X)\rightarrow\\{0,1\\}$ of monotonic
set functions such that $\alpha(H)=\beta(H)$ for all $H\in{\mathcal{E}}$, we
have

$\int_{X}fd\alpha=\int_{X}fd\beta.$

Proof. i) $\implies$ ii): The condition ii) is a particular case of condition
i). ii) $\implies$ i): let us first set, for any monotonic set function
$\alpha:{\mathcal{P}}(X)\rightarrow[0,+\infty]$ and for any $s\in(0,\infty)$
and $H\in{\mathcal{P}}(X)$, $\alpha_{s}(H)=1$ or $0$ if we have respectively
$\alpha(H)>s$ or $\alpha(H)\leqslant s$. If
$\alpha,\beta:{\mathcal{P}}(X)\rightarrow[0,+\infty]$ are two montonic set
functions such that $\alpha(H)=\beta(H)$ for all $H\in{\mathcal{E}}$, we also
have $\alpha_{s}(H)=\beta_{s}(H)$ for all $s\in(0,+\infty)$. According to ii),
we therefore have $\int_{X}fd\alpha_{s}=\int_{X}fd\beta_{s}$ for all
$s\in(0,+\infty)$. We then deduce that

$\int_{X}fd\alpha=\lim_{n}\sum_{i=1}^{\infty}\frac{1}{2^{n}}\int_{X}fd\alpha_{i/2^{n}}=\lim_{n}\sum_{i=1}^{\infty}\frac{1}{2^{n}}\int_{X}fd\beta_{i/2^{n}}=\int_{X}fd\beta.$

Lemma 2. Let $(X,{\mathcal{E}})$ be a paved set. If the function
$f\in[0,+\infty]^{X}$ is ${\mathcal{E}}$-measurable, then the functions
$\alpha f$, $f\wedge a$, $f\vee a-a$ are also ${\mathcal{E}}$-measurable for
all $a\in[0,+\infty)$.

Proof. For any monotonic set function $\alpha$ defined on ${\mathcal{P}}(X)$
with values in $\\{0,1\\}$ it follows that:

$\int_{X}(f\wedge a)d\alpha=\left(\int_{X}fd\alpha\right)\wedge a\mbox{ and
}\int_{X}(f\vee a-a)d\alpha=\left(\int_{X}fd\alpha\right)\vee a-a$

for any $a\in[0,+\infty)$ and $f\in[0,+\infty]^{X}$. We therefore deduce Lemma
2 from Lemma 1. $\quad\quad\quad\blacksquare$

Theorem 1. Let $(X,{\mathcal{E}})$ be a paved set and let
$f\in[0,+\infty]^{X}$. The following conditions are equivalent:

  1. i)

$f$ is ${\mathcal{E}}$-measurable,

  2. ii)

for each pair $a,b\in(0,+\infty)$ such that $a>b$, there exists a set
$H\in{\mathcal{E}}$ such that $\\{f\geqslant a\\}\subset H\subset\\{f>b\\}$,

  3. iii)

there exists a sequence
$\\{g_{n}\\}_{n\in{\mathbb{N}}}\subset{\mathcal{S}}^{+}(X,{\mathcal{E}})$ such
that $g_{n}\leqslant f$ for all $n\in{\mathbb{N}}$ and the sequence
$\\{g_{n}\wedge a\\}_{n\in{\mathbb{N}}}$ converges uniformly to $f\wedge a$
for all $a\in(0,+\infty)$.

Proof. i) $\implies$ ii): we reason by contradiction. If $f$ does not satisfy
condition ii), there exist two real numbers $a,b\in(0,+\infty)$ with $a>b$
such that

(1) $\nexists H\in{\mathcal{E}}\quad\mbox{such that }\\{f>a\\}\subset
H\subset\\{f>b\\}.$

Let us then define the two set functions
$\tau_{1},\tau_{2}:{\mathcal{P}}(X)\rightarrow[0,1]$ by

$\tau_{1}(A)=\left\\{\begin{array}[]{ll}0&\mbox{if }A=X\\\ \sup\\{g(x):x\notin
A\\}&\mbox{if }A\neq X\end{array}\right.$

and

$\tau_{2}(A)=\left\\{\begin{array}[]{ll}1&\mbox{if }A=\emptyset\\\
\inf\\{g(x):x\in A\\}&\mbox{if }A\neq\emptyset,\end{array}\right.$

where $g=(f\wedge a-f\wedge b)/(a-b)$. According to this definition of
$\tau_{1},\tau_{2}$, in virtue of (1) it follows that333Translator’s note: The
original has $\tau_{1}(H)<t$ in (3) and $\tau_{2}(H)>t$ in (4).

(2) $\displaystyle\tau_{1}(\emptyset)=\tau_{2}(\emptyset)=1\mbox{ and the
functions }\tau_{1},\tau_{2}\mbox{ are decreasing;}$ (3)
$\displaystyle\tau_{1}(H)\leqslant t\mbox{ and }\tau_{2}(H)=0,\mbox{ if
}H\in{\mathcal{E}},H\supset\\{g>t\\}\mbox{ and }t\in(0,1);$ (4)
$\displaystyle\tau_{1}(H)=1\mbox{ and }\tau_{2}(H)\geqslant t,\mbox{ if
}H\in{\mathcal{E}},H\subset\\{g>t\\}\mbox{ and }t\in(0,1).$

Now the two monotonic set functions
$\alpha,\beta:{\mathcal{P}}(X)\rightarrow[0,+\infty]$, thus defined

$\displaystyle\alpha(A)$
$\displaystyle=\sup\\{2-\tau_{1}(H)-\tau_{2}(H):H\in{\mathcal{E}}\mbox{ and
}H\subset A\\},$ $\displaystyle\beta(A)$
$\displaystyle=\inf\\{2-\tau_{1}(H)-\tau_{2}(H):H\in{\mathcal{E}}\mbox{ and
}H\supset A\\},$

are therefore equal on ${\mathcal{E}}$ and $\beta\geqslant\alpha$, according
to (2). Moreover, we have $\beta\\{g>t\\}>1>\alpha\\{g>t\\}$ for all
$t\in(0,1)$, according to (3) and (4); this proves that
$\int_{X}gd\beta>\int_{X}gd\alpha$, that is the function $g=(f\wedge a-f\wedge
b)/(a-b)$ is not ${\mathcal{E}}$-measurable, according to the definition of
${\mathcal{E}}$-measurability. But this contradicts the hypothesis that $f$ is
${\mathcal{E}}$-measurable; indeed from the ${\mathcal{E}}$-measurability of
$f$ we must deduce that $g$ must be ${\mathcal{E}}$-measurable, according to
Lemma 2. We thus complete the proof of i) $\implies$ ii).

ii) $\implies$ iii): if $f\in[0,+\infty]^{X}$ satisfies condition ii), for any
$i,n\in{\mathbb{N}}$ there exists a set $H_{i,n}\in{\mathcal{E}}$ such that

$\left\\{f\geqslant\frac{i+1}{2^{n}}\right\\}\subset
H_{i,n}\subset\left\\{f>\frac{i}{2^{n}}\right\\}\quad\mbox{and}\quad
H_{i+1,n}\subset H_{i,n}$

We therefore have

$f\geqslant\frac{1}{2^{n}}\sum_{i=1}^{\infty}\varphi_{H_{i,n}}\geqslant
f\wedge\frac{1}{2^{n-1}}-\frac{1}{2^{n-1}}.$

The sequence $\\{g_{n}\\}_{n\in{\mathbb{N}}}$, where
$g_{n}=1/2^{n}\sum_{i=1}^{\infty}\varphi_{H_{i,n}}$, satisfies condition iii).

iii) $\implies$ i): the ${\mathcal{E}}$-measurability of all
$g\in{\mathcal{S}}^{+}(X,{\mathcal{E}})$ and the properties of the integral
$\int_{X}\mbox{--- }d\alpha$ (see [3], Prop. 1) entail the
${\mathcal{E}}$-measurability of $f$. $\quad\quad\quad\blacksquare$

Remark 1. Let $(X,{\mathcal{E}})$ be a paved set, let
$\delta:{\mathcal{E}}\rightarrow[0,+\infty]$ be a set function such that
$\delta(\emptyset)=0$ and $\delta(A)\leqslant\delta(B)$ for all
$A,B\in{\mathcal{E}}$ with $A\subset B$. The preceding theorem makes it
possible to define in a satisfying manner the integral $\int_{X}\mbox{---
}d\delta$; it is defined by setting $\int_{X}fd\delta=\int_{X}fd\alpha$, where
$\alpha:{\mathcal{P}}(X)\rightarrow[0,+\infty]$ is a monotonic set function
such that $\alpha(H)=\delta(H)$ for all $H\in{\mathcal{E}}$, when
$f\in{\mathcal{M}}(X,{\mathcal{E}})$ and $\int_{X}f^{+}d\alpha<+\infty$ or
$\int_{X}f^{-}d\alpha<+\infty$. According to this definition of the integral
we can analyse the properties of $\delta$ by means of the properties of
$\int_{X}\mbox{--- }d\alpha$. For example let $(X,{\mathcal{E}})$ be a paved
set that is stable for $(\cap f,\cup f)$; we will see in Proposition 2 that
the family of ${\mathcal{E}}$-measurable functions
${\mathcal{M}}(X,{\mathcal{E}})$ is a convex cone of functions stable for
$(\wedge f,\vee f)$. The following conditions are equivalent

  1. i)

$\delta(A\cap B)+\delta(A\cup B)=\delta(A)+\delta(B),\forall
A,B\in{\mathcal{E}}$,

  2. ii)

$\int_{X}(f+g)d\delta=\int_{X}fd\delta+\int_{X}gd\delta,\forall
f,g\in{\mathcal{M}}^{+}(X,{\mathcal{E}})$,

  3. iii)

$\int_{X}(f\wedge g)d\delta+\int_{X}(f\vee
g)d\delta=\int_{X}fd\delta+\int_{X}gd\delta,\forall
f,g\in{\mathcal{M}}^{+}(X,{\mathcal{E}})$.

If in i), ii) and iii) we change “$=$” to “$\leqslant$” or “$\geqslant$”, we
again obtain the equivalence of i), ii) and iii) (see [4]).

##  2. Properties of ${\mathcal{E}}$-measurable functions.

From the definition of ${\mathcal{E}}$-measurability and Property ii) of
Theorem 1, we obtain the following propositions on the properties of
${\mathcal{E}}$-measurable functions.

Proposition 1. Let $(X,{\mathcal{E}})$ be a paved set and let
$\psi:[0,+\infty]\rightarrow[0,+\infty]$ be a continuous and increasing
function. We have

  1. i)

if $\psi(0)=0$, $f\in{\mathcal{M}}^{+}(X,{\mathcal{E}})$, then $\psi\circ
f\in{\mathcal{M}}^{+}(X,{\mathcal{E}})$ (note we can dispense with the
equality $\psi(0)=0$ if $X\in{\mathcal{E}}$);

  2. ii)

if the sequence
$\\{f_{n}\\}_{n\in{\mathbb{N}}}\subset{\mathcal{M}}^{+}(X,{\mathcal{E}})$
converges uniformly to $f$, then $f\in{\mathcal{M}}^{+}(X,{\mathcal{E}})$.

Proposition 2. Let $(X,{\mathcal{E}})$ be a paved set. Then

  1. i)

for ${\mathcal{M}}^{+}(X,{\mathcal{E}})$ to be stable for $(\wedge f)$
(respectively, for $(\vee f)$) it is necessary and sufficient that
${\mathcal{E}}$ be stable for $(\cap f)$ (respectively, for $(\cup f)$);

  2. ii)

to have $f+g\in{\mathcal{M}}^{+}(X,{\mathcal{E}})$ for any
$f,g\in{\mathcal{M}}^{+}(X,{\mathcal{E}})$, it is necessary and sufficient
that ${\mathcal{E}}$ be stable for $(\cap f,\cup f)$;

  3. iii)

for ${\mathcal{M}}^{+}(X,{\mathcal{E}})$ to be stable for $(\wedge d)$
(respectively, for $(\vee d)$) it is necessary and sufficient that
${\mathcal{E}}$ be stable for $(\cap d)$ (respectively, for $(\cup d)$); in
that case, for $f\in[0,+\infty]$ to be ${\mathcal{E}}$-measurable it is
necessary and sufficient that the set $\\{f\geqslant t\\}\in{\mathcal{E}}$
(respectively, $\\{f>t\\}\in{\mathcal{E}}$) for all $t\in(0,+\infty)$.

Proof. We are only going to show property ii). If $f+g$ is ${\mathcal{E}}$
measurable for all pairs of functions
$f,g\in{\mathcal{M}}^{+}(X,{\mathcal{E}})$, we have that
$\varphi_{A}+\varphi_{B}$ is ${\mathcal{E}}$ measurable for all
$A,B\in{\mathcal{E}}$; it follows therefore that $A\cap B$ and $A\cup
B\in{\mathcal{E}}$, according to ii) of Theorem 1; that is ${\mathcal{E}}$ is
stable for $(\cap f,\cup f)$. On the other hand, suppose that ${\mathcal{E}}$
is stable for $(\cap f,\cup f)$, let us verify that
${\mathcal{M}}^{+}(X,{\mathcal{E}})$ is stable for summation. If
$g_{1}=\sum_{i=1}^{n}a_{i}\varphi_{A_{i}}$ and
$g_{2}=\sum_{i=1}^{n}b_{i}\varphi_{B_{i}}$ belong to
${\mathcal{S}}^{+}(X,{\mathcal{E}})$, the function $g_{1}+g_{2}$ belongs to
${\mathcal{S}}^{+}(X,{\mathcal{E}})$, because we have for all
$t\in(0,+\infty)$ the equality

$\\{g_{1}+g_{2}>t\\}=\cup_{(a,b)\in Z}(\\{g_{1}\geqslant
a\\}\cap\\{g_{2}\geqslant b\\}),$

where $Z$ is the finite set

$\\{(g_{1}(x),g_{2}(x))\in{\mathbb{R}}^{2}:x\in X\mbox{ and
}g_{1}(x)+g_{2}(x)>t\\}.$

If $f,g\in{\mathcal{M}}^{+}(X,{\mathcal{E}})$ are bounded, there exist two
sequences of bounded ${\mathcal{E}}$-measurable functions
$\\{g_{n}\\}_{n\in{\mathbb{N}}}$, $\\{f_{n}\\}_{n\in{\mathbb{N}}}$, which
converge uniformly, respectively, to $g$ and $f$ (see Th. 1, property iii));
since
$\\{g_{n}+f_{n}\\}_{n\in{\mathbb{N}}}\subset{\mathcal{S}}^{+}(X,{\mathcal{E}})$
converges uniformly to $g+f$, we therefore have that
$f+g\in{\mathcal{M}}^{+}(X,{\mathcal{E}})$, by property ii) of Proposition 1.
Finally, taking into account the equality
$\int_{X}(f+g)d\alpha=\lim_{n}\int_{X}(f\wedge n+g\wedge n)d\alpha$, true for
any $f,g\in[0,+\infty]^{X}$ and for any monotonic set function
$\alpha:{\mathcal{P}}(X)\rightarrow[0,+\infty]$, it follows then that
$f+g\in{\mathcal{M}}^{+}(X,{\mathcal{E}})$, also if
$f,g\in{\mathcal{M}}^{+}(X,{\mathcal{E}})$ are not bounded.
$\quad\quad\quad\blacksquare$

In particular cases, the class ${\mathcal{M}}^{+}(X,{\mathcal{E}})$ has new
properties. For example, if $X$ is a topological space in which every
countable open cover of $X$ contains a finite open cover of $X$, any function
$f\in[0,+\infty)^{X}$ that is measurable with respect to the paving of closed
sets of $X$ (that is, $f\in[0,+\infty)^{X}$ is upper semi-continuous) is
bounded; or if $X$ is a pseudocompact topological space (that is, every
continuous function $g:X\rightarrow{\mathbb{R}}$ is bounded), any function
$f\in[0,\infty)^{X}$ that is measurable with respect to the paving comprised
of the sets of the form $\\{g=0\\}$, where $g:X\rightarrow{\mathbb{R}}$ is
continuous, is also bounded. This depends on the semi-compactness of the
pavings in question (see [8]); recall that the paving ${\mathcal{E}}$ on $X$
is said to be semi-compact (see [5]), if
$\\{H_{n}\\}_{n\in{\mathbb{N}}}\subset{\mathcal{E}}$ and
$\cap_{n=1}^{\infty}H_{n}=\emptyset$ entails that there exists
$n_{0}\in{\mathbb{N}}$ such that $\cap_{n=1}^{n_{0}}H_{n}=\emptyset$.

Theorem 2. Let $X$ be a set endowed with a paving ${\mathcal{E}}$ that is
stable for $(\cap f)$. The following conditions are equivalent:

  1. i)

${\mathcal{E}}$ is semi-compact,

  2. ii)

any function $f\in{\mathcal{M}}^{+}(X,{\mathcal{E}})\cap{\mathbb{R}}^{X}$ is
bounded,

  3. iii)

any function $f\in{\mathcal{M}}^{+}(X,{\mathcal{E}})$ has a maximum,

  4. iv)

any sequence $\\{f_{n}\\}_{n\in{\mathbb{N}}}$, decreasing to zero, converges
uniformly to zero, if all $f_{n}$ are ${\mathcal{E}}$-measurable,

  5. v)

for any sequence
$\\{f_{n}\\}_{n\in{\mathbb{N}}}\subset{\mathcal{M}}^{+}(X,{\mathcal{E}})$,
decreasing to zero, and for any monotonic
$\alpha:{\mathcal{P}}(X)\rightarrow[0,+\infty]$ such that
$\int_{X}f_{1}d\alpha<+\infty$, we have $\lim_{n}\int_{X}f_{n}d\alpha=0$.

Proof. It suffices to suitably adapt the proof of an analogous theorem of I.
Glicksberg [2]. i) $\implies$ iii) by property ii) of Theorem 1, there exist
sets $H_{n}\in{\mathcal{E}}$ such that $\\{f\geqslant a_{n+1}\\}\subset
H_{n}\subset\\{f>a_{n}\\}$, for all $n\in{\mathbb{N}}$, if
$f\in{\mathcal{M}}^{+}(X,{\mathcal{E}})$,
$\\{a_{n}\\}_{n\in{\mathbb{N}}}\subset(0,+\infty)$, $a_{n+1}>a_{n}$ for all
$n\in{\mathbb{N}}$ and $\lim_{n}a_{n}=\sup\\{f(x):x\in X\\}$ (we assume that
$f\neq 0$); since $\cap_{n=1}^{m}H_{n}\supset\\{f\geqslant
a_{m+1}\\}\neq\emptyset$ for all $m\in{\mathbb{N}}$, we deduce from the semi-
compactness of ${\mathcal{E}}$ that $\cap_{n=1}^{\infty}H_{n}\neq\emptyset$;
if $x_{0}\in\cap_{n=1}^{\infty}H_{n}$, we therefore have $f(x_{0})\geqslant
a_{n}$ for all $n\in{\mathbb{N}}$, that is $f(x_{0})=\sup(f)$.

iii) $\implies$ ii): is an immediate consequence of iii).

ii) $\implies$ i): let $\\{H_{n}\\}_{n\in{\mathbb{N}}}\subset{\mathcal{E}}$
and $\cap_{n=1}^{\infty}H_{n}=\emptyset$. The sets $G_{n}=\cap_{i=1}^{n}H_{n}$
belong to ${\mathcal{E}}$ and the function
$\sum_{n=1}^{\infty}n\varphi_{G_{n}}\in{\mathcal{M}}^{+}(X,{\mathcal{E}})\cap{\mathbb{R}}^{X}$;
by hypothesis ii) the function $\sum_{n=1}^{\infty}n\varphi_{G_{n}}$ is
bounded; then there exists a number $n_{0}\in{\mathbb{N}}$ such that
$G_{n_{0}}=\emptyset$, that is $\cap_{n=1}^{n_{0}}H_{n}=\emptyset$.

i) $\implies$ iv): let
$\\{f_{n}\\}_{n\in{\mathbb{N}}}\subset{\mathcal{M}}^{+}(X,{\mathcal{E}})$ be a
sequence decreasing to zero. Let us show that $\\{f_{n}\\}_{n}$ converges to
zero uniformly, that is for any real number $a>0$ there exists
$n_{0}\in{\mathbb{N}}$ such that $f_{n_{0}}\leqslant a$. We choose a sequence
$\\{a_{n}\\}_{n}\subset(0,+\infty)$ such that $a_{n}<a$ and $\lim_{n}a_{n}=a$.
By the measurability of all $f_{n}$ and property ii) of Theorem 1, there
exists for all $n\in{\mathbb{N}}$ a set $H_{n}\in{\mathcal{E}}$ such that
$\\{f_{n}\geqslant a\\}\subset H_{n}\subset\\{f_{n}>a_{n}\\}$. We therefore
have $\cap_{n=1}^{\infty}H_{n}=\emptyset$, because $\lim_{n}f_{n}=0$; we then
deduce that there exists $n_{0}\in{\mathbb{N}}$ such that
$\cap_{n=1}^{n_{0}}H_{n}=\emptyset$, by hypothesis i). Therefore
$\\{f_{n_{0}}\geqslant a\\}\subset\cap_{n=1}^{n_{0}}H_{n}=\emptyset$, that is
the sequence $\\{f_{n}\\}_{n\in{\mathbb{N}}}$ converges uniformly to zero.

iv) $\implies$ v): this is a consequence of the properties of the integral
$\int_{X}\mbox{--- }d\alpha$ (see [3], Prop. 1, property (4)).

v) $\implies$ i): it suffices to demonstrate that there exists
$n_{0}\in{\mathbb{N}}$ such that $H_{n_{0}}=\emptyset$, if
$\\{H_{n}\\}_{n\in{\mathbb{N}}}\subset{\mathcal{E}}$ with
$\cap_{n=1}^{\infty}H_{n}=\emptyset$ and $H_{n+1}\subset H_{n}$ for all
$n\in{\mathbb{N}}$. We reason by contradiction; if $H_{n}\neq\emptyset$ for
all $n\in{\mathbb{N}}$, the monotone set function
$\alpha_{{\mathcal{F}}}:{\mathcal{P}}(X)\rightarrow\\{0,1\\}$ that is equal to
1 only for those sets of the filter ${\mathcal{F}}$ generated by the $H_{n}$,
is such that $\alpha_{{\mathcal{F}}}=1$ for all $n\in{\mathbb{N}}$; which is
contrary to hypothesis v), since the sequence
$\\{\varphi_{H_{n}}\\}_{n\in{\mathbb{N}}}$ decreases to zero.
$\quad\quad\quad\blacksquare$

When the paving ${\mathcal{E}}$ on $X$ is an algebra of sets we have the
following characterisation of ${\mathcal{E}}$-measurability.

Theorem 3. Let $(X,{\mathcal{A}})$ be a paved set, ${\mathcal{A}}$ an algebra
of sets, $f\in\overline{{\mathbb{R}}}^{x}$. The following conditions are
equivalent:

  1. i)

$f$ is ${\mathcal{A}}$-measurable,

  2. ii)

for any ultrafilter ${\mathcal{F}}$ of the algebra ${\mathcal{A}}$ there
exists the limit $\lim_{{\mathcal{F}}}f\in\overline{{\mathbb{R}}}$,

  3. iii)

for any real number $a>0$, there exists a finite partition
$\\{H_{i}\\}_{i=1}^{n}\subset{\mathcal{A}}$ of $X$ such that for all $i$ we
have

$f(x)\leqslant a+f(y)\quad\mbox{if }x,y\in H_{i}$

or

$f(x)\geqslant\frac{1}{a}\quad\mbox{if }x\in H_{i}$

or

$f(x)\leqslant-\frac{1}{a}\quad\mbox{if }x\in H_{i}.$

Proof. i) $\implies$ ii): we can limit ourselves to verifying that for all
$f\in[0,+\infty]^{X}\cap{\mathcal{M}}(X,{\mathcal{E}})$ there exists the limit
$\lim_{{\mathcal{F}}}f\in\overline{{\mathbb{R}}}$. Let
${\mathcal{F}}^{\prime}\subset{\mathcal{P}}(X)$ be the filter of
${\mathcal{P}}(X)$ generated by the ultrafilter ${\mathcal{F}}$ of
${\mathcal{A}}$, and let $\alpha_{{\mathcal{F}}^{\prime}}$ and
$\beta_{{\mathcal{F}}^{\prime}}$ be the monotonic set functions defined in the
introduction. For these set functions we have
$\int_{X}fd\alpha_{{\mathcal{F}}^{\prime}}=\liminf_{{\mathcal{F}}}f$ and
$\int_{X}fd\beta_{{\mathcal{F}}^{\prime}}=\limsup_{{\mathcal{F}}}f$; since
$\alpha_{{\mathcal{F}}^{\prime}}(H)=\beta_{{\mathcal{F}}^{\prime}}(H)$ for any
$H\in{\mathcal{A}}$, it follows that
$\limsup_{{\mathcal{F}}}f=\liminf_{{\mathcal{F}}}f$, by the definition of
${\mathcal{A}}$-measurability; this proves that $\lim_{{\mathcal{F}}}f$ exists
and belongs to $\overline{{\mathbb{R}}}$.

ii) $\implies$ iii): according to ii), for any ultrafilter ${\mathcal{F}}$ of
the algebra ${\mathcal{A}}$ there exists a set
$H_{{\mathcal{F}}}\in{\mathcal{F}}$444Translator’s note: The original has
$H_{{\mathcal{F}}}\in{\mathcal{A}}$. such that $f(x)-f(y)\leqslant a$ or
$f(x)\geqslant 1/a$ or else $f(x)\leqslant-1/a$ for $x,y\in
H_{{\mathcal{F}}}$. The family $\\{X-H_{{\mathcal{F}}}:{\mathcal{F}}\mbox{ an
ultrafilter of }{\mathcal{A}}\\}$ cannot be contained in any ultrafilter of
${\mathcal{A}}$; there exists therefore a finite number
${\mathcal{F}}_{1},{\mathcal{F}}_{2},\ldots,{\mathcal{F}}_{n}$ of ultrafilters
of ${\mathcal{A}}$ such that

$\cap_{i=1}^{n}(X-H_{{\mathcal{F}}_{i}})=\emptyset;$

the proof of the implication ii) $\implies$ iii) is thus complete.

iii) $\implies$ i): it suffices to verify this implication when
$f\in[0,+\infty]^{X}$ is bounded. In that case the construction of a sequence
$\\{g_{n}\\}_{n}\subset{\mathcal{M}}^{+}(X,{\mathcal{A}})$ such that
$\\{g_{n}\\}_{n}$ converges uniformly to the function $f$ is immediate; thus
we can deduce that $f$ is ${\mathcal{A}}$-measurable, by the properties of
${\mathcal{A}}$-measurable functions. $\blacksquare$

Remark 2. Let $X={\mathbb{R}}$ and let ${\mathcal{A}}_{{\mathbb{R}}}$ be the
algebra generated by the bounded and unbounded intervals of ${\mathbb{R}}$.
For all $x\in{\mathbb{R}}$, let ${\mathcal{F}}_{x}$, ${\mathcal{F}}_{x^{+}}$,
${\mathcal{F}}_{x^{-}}$, ${\mathcal{F}}_{+\infty}$, ${\mathcal{F}}_{-\infty}$
be the ultrafilters of ${\mathcal{A}}_{{\mathbb{R}}}$ generated, respectively,
by the set $\\{x\\}$, by the intervals $(x,x+a)$, where $a\in(0,+\infty)$, by
the intervals $(x-a,x)$, where $a\in(0,+\infty)$, by the intervals
$(a,+\infty)$, where $a\in{\mathbb{R}}$, and by the intervals $(-\infty,a)$,
where $a\in{\mathbb{R}}$; all the ultrafilters of
${\mathcal{A}}_{{\mathbb{R}}}$ are of these types. According to Theorem 3, it
follows that a function $f\in\overline{{\mathbb{R}}}^{X}$ is
${\mathcal{A}}_{{\mathbb{R}}}$-measurable if and only if the following limits
exist: $\lim_{y\rightarrow x^{+}}f(y)$, $\lim_{y\rightarrow x^{-}}f(y)$,
$\lim_{y\rightarrow+\infty}f(y)$, and $\lim_{y\rightarrow-\infty}f(y)$; that
is $f$ is devoid of oscillatory discontinuities. This example can easily be
adapted when ${\mathcal{A}}$ is the algebra generated by the intervals that
are semi-open on the right (or on the left), and when $X={\mathbb{R}}^{n}$ and
${\mathcal{A}}$ is the algebra generated by the $n$-dimensional intervals.

Remark 3. Theorem 3 can be presented in a more suggestive manner. If we denote
by $[{\mathcal{A}}]$ the Stone space (the space of ultrafilters of
${\mathcal{A}}$)(see [6]), associated with the algebra ${\mathcal{A}}$, for a
function $f\in\overline{{\mathbb{R}}}^{X}$ to be ${\mathcal{A}}$-measurable it
is necessary and sufficient that there exists a continuous function
$\overline{f}:[{\mathcal{A}}]\rightarrow\overline{{\mathbb{R}}}$ such that
$\overline{f}({\mathcal{F}}_{x})=f(x)$ for any ultrafilter
${\mathcal{F}}_{x}=\\{H\in{\mathcal{A}}:x\in H\\}$, where $x\in X$. The class
of bounded and ${\mathcal{A}}$-measurable functions is therefore an algebra of
functions, isomorphic to the algebra of continuous functions from
$[{\mathcal{A}}]$ to ${\mathbb{R}}$.

Remark 4. In the particular case where the algebra ${\mathcal{A}}$ is a
$\sigma$-algebra, we have that $f+g\in{\mathcal{M}}(X,{\mathcal{A}})$ if $f+g$
is defined over $X$ and $f,g\in{\mathcal{M}}(X,{\mathcal{E}})$. This property
is not true if ${\mathcal{A}}$ is only an algebra. Indeed if we consider the
algebra ${\mathcal{A}}\subset{\mathcal{P}}({\mathbb{N}})$ of finite and
cofinite subsets of ${\mathbb{N}}$, for a function
$f:{\mathbb{N}}\rightarrow\overline{{\mathbb{R}}}$ to be
${\mathcal{A}}$-measurable it is necessary and sufficient that there exists
the limit $\lim_{n}f(n)$; in that case we can therefore easily choose two
functions $f,g:{\mathbb{N}}\rightarrow\overline{{\mathbb{R}}}$ such that $f+g$
is defined, but the function $f+g$ is not ${\mathcal{A}}$-measurable.

Remark 5. Let $\mu:{\mathcal{P}}(X)\rightarrow[0,+\infty]$ be a monotonic or
non-monotonic set function with $\mu(\emptyset)=0$. We know that the family of
sets that are $\mu$-measurable in the sense of Carathéodory is a set algebra
${\mathcal{A}}_{\mu}\subset{\mathcal{P}}(X)$. Let
$\delta:{\mathcal{A}}_{\mu}\rightarrow[0,+\infty]$ be the monotone set
function defined by $\delta(H)=\mu(H)$ for all $H\in{\mathcal{A}}_{\mu}$; a
function $f$ is said to be $\mu$-measurable if $f$ is
${\mathcal{A}}_{\mu}$-measurable. For these $\mu$-measurable functions we have
linearity, that is $\int(f+g)d\delta=\int_{X}fd\delta+\int_{X}gd\delta$ (see
Remark 1 for the definition of the integral $\int_{X}\mbox{--- }d\delta$) for
all $f,g\in{\mathcal{M}}^{+}(X,{\mathcal{A}}_{\mu})$. Furthermore we have that
the sets $\\{t\in{\mathbb{R}}:\\{f>t\\}\notin{\mathcal{A}}_{\mu}\\}$ and
$\\{t\in{\mathbb{R}}:\\{f\geqslant t\\}\notin{\mathcal{A}}_{\mu}\\}$ are
countable, if $f\in{\mathcal{M}}(X,{\mathcal{A}}_{\mu})$ and
$\int_{X}|f|d\delta<+\infty$. The class of functions $f\in{\mathbb{R}}^{X}$
that are $\mu$-measurable, such that $\int_{X}|f|d\delta<+\infty$, is a Riesz
space (see [4]).

##  3. Measurable functions with respect to two pavings.

In this paragraph we consider two pavings ${\mathcal{K}},{\mathcal{U}}$ on
$X$, both stable for $(\cap f,\cup f)$; we define on ${\mathcal{P}}(X)$ the
relation “$\ll$” in this manner: “$A\ll B$ if and only if there exists a pair
$(K,U)\in{\mathcal{K}}\times{\mathcal{U}}$ such that $A\subset K\subset
U\subset B$”. Let $D$ be the set of dyadic numbers in the interval $[0,1]$;
let $\\{H_{t}\\}_{t\in D}\subset{\mathcal{P}}(X)$ be a family of sets such
that $H_{0}=X$ and $H_{t}\gg H_{s}$ for all $t,s\in D$ with $t<s$; then the
function $f:X\rightarrow[0,1]$, defined by $f(x)=\sup\\{t\in D:x\in H_{t}\\}$,
is either ${\mathcal{K}}$-measurable or ${\mathcal{U}}$-measurable. Indeed for
any $a,b\in(0,+\infty)$ with $a>b$, let $t$ and $r$ be two numbers $\in D$
such that $a>t>r>b$, it therefore follows that $\\{f\geqslant a\\}\subset
H_{t}\ll H_{r}\subset\\{f>b\\}$; since there exists a pair
$(K,U)\in{\mathcal{K}}\times{\mathcal{U}}$ such that $H_{t}\subset K\subset
U\subset H_{r}$, we finally have that $\\{f\geqslant a\\}\subset K\subset
U\subset\\{f>b\\}$, that is $f$ is ${\mathcal{K}}$-measurable and
${\mathcal{U}}$-measurable.

We say that the pair of pavings $({\mathcal{K}},{\mathcal{U}})$ satisfies
property ($N$) if for any pair $(K,U)\in{\mathcal{K}}\times{\mathcal{U}}$ with
$K\subset U$, there exists a pair
$(K^{\prime},U^{\prime})\in{\mathcal{K}}\times{\mathcal{U}}$ such that
$K\subset U^{\prime}\subset K^{\prime}\subset U$.

Lemma 3. Let $k,u\in[0,+\infty]^{X}$, and let $({\mathcal{K}},{\mathcal{U}})$
be a pair of pavings on $X$ that satisfy property ($N$). If $k\leqslant u$ and
if $\\{k\geqslant t\\}\in{\mathcal{K}}$ and $\\{u>t\\}\in{\mathcal{U}}$ for
any $t\in(0,+\infty)$, then there exists a function
$f\in{\mathcal{M}}^{+}(X,{\mathcal{K}})\cap{\mathcal{M}}^{+}(X,{\mathcal{U}})$
such that $k\leqslant f\leqslant u$.

Proof. By property i) of Proposition 1 we can suppose that $u\leq 1$. For any
$t\in D$, where $D$ is the set of dyadic numbers in the interval $[0,1]$, we
set $K_{t}=\\{k\geqslant t\\}$ and $U_{t}=\\{u>t\\}$. We construct a family of
sets $\\{F_{t}\\}_{t\in D}\subset{\mathcal{P}}(X)$, enjoying the following
properties555Translator’s note: In (2), the original has $G_{t}$ in place of
$U_{t}$, and $H_{r}$ in place of $K_{r}$.:

(1) $\displaystyle F_{0}$ $\displaystyle=$ $\displaystyle X,\quad
F_{1}=\emptyset,$ (2) $\displaystyle F_{r}$ $\displaystyle\ll$ $\displaystyle
F_{t},\quad F_{r}\ll U_{t},\quad K_{r}\ll F_{t},\quad\mbox{if }t,r\in D\mbox{
and }t<r;$

in fact if we suppose sets $F_{t}$ satisfying (1) and (2) have been defined
for all $t=i/2^{n}$, where $0\leqslant i\leqslant 2^{n}$, then we can choose a
set $F_{t_{0}}$ for $t_{0}=(2i+1)/2^{n+1}$, where $0\leqslant i\leqslant
2^{n}-1$, satisfying the relations: $F_{i/2^{n}}\gg F_{t_{0}}$, $F_{t_{0}}\gg
F_{(i+1)/2^{n}}$, $U_{i/2^{n}}\gg F_{t_{0}}$, $F_{t_{0}}\gg K_{(i+1)/2^{n}}$,
according to the relations (true by hypothesis or by construction):

$F_{i/2^{n}}\gg F_{(i+1)/2^{n}},\;F_{i/2^{n}}\gg
K_{(i+1)/2^{n}},\;U_{i/2^{n}}\gg F_{(i=1)/2^{n}},\;U_{i/2^{n}}\gg
K_{(i+1)/2^{n}}.$

The function $f:X\rightarrow[0,+\infty]$, defined by $f(x)=\sup\\{t\in D:x\in
F_{t}\\}$, is thus ${\mathcal{K}}$-measurable and ${\mathcal{U}}$-measurable
and $k\leqslant f\leqslant u$. $\quad\quad\quad\blacksquare$

By Lemma 3 and property iii) of Proposition 2 we deduce the following
theorems:

Theorem 4. Let ${\mathcal{K}}$ and ${\mathcal{U}}$ be two pavings on $X$,
stable for $(\cap f,\cup f)$. The following properties are equivalent:

  1. i)

the pair $({\mathcal{K}},{\mathcal{U}})$ satisfies property ($N$),

  2. ii)

for any pair $(K,U)\in{\mathcal{K}}\times{\mathcal{U}}$ there exists a
function
$f\in{\mathcal{M}}^{+}(X,{\mathcal{K}})\cap{\mathcal{M}}^{+}(X,{\mathcal{U}})$
such that $\varphi_{K}\leqslant f\leqslant\varphi_{U}$, if $K\subset U$.
$\quad\quad\quad\blacksquare$

Theorem 5. Let ${\mathcal{K}}$ be a paving stable for $(\cap d,\cup f)$ and
let ${\mathcal{U}}$ be a paving stable for $(\cap f,\cup d)$. The following
properties are equivalent:

  1. i)

the pair $({\mathcal{K}},{\mathcal{U}})$ satisfies property ($N$),

  2. ii)

for any
$(k,u)\in{\mathcal{M}}^{+}(X,{\mathcal{K}})\times{\mathcal{M}}^{+}(X,{\mathcal{U}})$
there exists a function
$f\in{\mathcal{M}}^{+}(X,{\mathcal{K}})\cap{\mathcal{M}}^{+}(X,{\mathcal{U}})$
such that $k\leqslant f\leqslant u$, if $k\leqslant u$.
$\quad\quad\quad\blacksquare$

Let $X$ be a topological space and let ${\mathcal{K}}$ (respectively,
${\mathcal{U}}$) be a paving on $X$ comprised of closed sets (respectively,
open sets); the pair of pavings $({\mathcal{K}},{\mathcal{U}})$ satisfies
property ($N$) if and only if $X$ is normal. In that case Theorems 4 and 5 are
well known theorems on normal spaces (see [7]).

## BIBLIOGRAPHY

  * [1] E. De Giorgi, G. Letta, Une notion général de convergence faible pour des fonctions croissantes d’ensemble, Ann. Sc. Nor. Pisa, 4 (IV) (1977), pp. 61–99.
  * [2] I. Glicksberg, The representation of functionals by integrals, Duke Math. J., 19 (1952), pp. 253–261.
  * [3] G. H. Greco, Integrale monotono, Rend. Sem. Mat. Padova, 57 (1977), pp. 149–166.
  * [4] G. H. Greco, R. C. Bassanezi, Sull’additività di alcuni funzionali definiti su coni di funzioni (preprint)666Translator’s note: The intended reference is clearly R. C. Bassanezi, G. H. Greco, Sull’additività dell’integrale, Rend. Sem. Mat. Padova, 72 (1984), pp. 249–275..
  * [5] P. A. Meyer, Probabilités et potentiel, Hermann, 1966.
  * [6] R. Sikorski, Boolean algebras, Springer-Verlag, 1969.
  * [7] H. Tong, Some characterizations of normal and perfectly normal spaces, Duke Math. J., 19 (1952), pp. 289–292.
  * [8] V. S. Varadarayan, Measures on topological spaces, Amer. Math. Soc. Transl., ser. II, 48 (1965), pp. 161–228.


# The dynamic three-dimensional Anderson localization of optical fields in
active percolating systems

Gennadiy Burlak  Centro de Investigación en Ingeniería y Ciencias Aplicadas,
Universidad Autónoma del Estado de Morelos, Cuernavaca, Mor., México

###### Abstract

We study three-dimensional optical Anderson localization in medium with a
percolating disorder, where the percolating clusters are filled by the light
nanoemitters in the excited state. The peculiarity of situation is that in
such materials the field clusters join a fractal radiating system where the
light is both emitted and scattered by the inhomogeneity of clusters. Our
numerical FDTD simulations found that in this nonlinear compound the number of
localized field bunches drastically increases after the lasing starts. At
longer times the bunches leave nonlinear percolating structure and are
radiated from the medium. Monitoring the output of system allows to observe
experimentally such dynamic localized bunches in three-dimensional setup.

Introduction. Disordered photonic materials can diffuse and localize light
through random multiple scattering that leads to formation of the
electromagnetic modes depending on the structural correlations, scattering
strength, and dimensionality of the system Riboli:2014 , Vinck:2011 ,
Sheng:2010 , Wang:2011 , jendrzejewski:2012 , Segev:2013a , Wiersma:2013a .
The Anderson localization was predicted as a non-interacting linear
interference effect anderson:1958 . However, in real systems the non-
negligible interactions between light and medium can take place. Therefore
important aspect of the optical localization is the interplay between
nonlinear interactions and linear Anderson effect Segev:2013a . Nonlinear
interactions appear in optics, due to nonlinear responses of a disordered
medium that normally gives rise to indirect interactions between the photons
through various mechanisms. In case of classical waves the localization can be
interpreted as interference between the various amplitudes associated with the
scattering paths of optical waves propagating among the diffusers. The study
for Anderson transition in 3-D optical systems still has not been conclusive
despite considerable efforts. The localization transition may be difficult to
reach for the light waves due to various effects in dense disordered media
required to achieve strong scattering [see Skipetrov:2016a  and references
therein]. The experimental observation of optical Anderson localization
Sanli:2009a  just below the Anderson transition in 3D disordered medium shows
strong fluctuations of the wave function that leads to nontrivial length-scale
dependence of the intensity distribution (multifractality).

Here we study the optical Anderson localization in 3D percolating disorder,
where the percolating clusters are filled by the light nanoemitters in the
excited state. The peculiarity of situation is that in such materials the
field clusters join a fractal nonlinear radiating system where the light is
both emitted and scattered by the inhomogeneity of clusters. Such system may
be suggested as an extension to 3D optical case the localization released in
one-dimensional waveguide in the presence of a controlled disorder Billy:2008a
. In 3D disordered percolating systems the optical transport was observed
Burlak:2009a  and also it is found that the random lasing assisted by
nanoemitters incorporated into such a disordered structure can occur
burlak:2015 . The system with spanning clusters produces a global percolation
that results in qualitatively modification of its spatial properties. One can
argues that already in a vicinity of the percolating phase transition the
fractal dimension of such system $D_{H}\simeq 2.54$ considerably differs from
the dimension of the embedded space $D=3$ (multifractality). The crucial
question is whether the optical Anderson localization can be achieved for a
non-integer dimension case with a fractal (Hausdorff) dimension of $D_{H}<3$,
where the strong randomness for properties of the system is expected. In this
paper we show that the dynamic field localization can arise in such 3D active
fractal percolation system. To the best of our knowledge, the dynamic 3D
Anderson localization assisted by fractal disordered inhomogeneity is
discussed here for the first time.

Basic equations. The percolation normally refers to the leakage of fluid or
gas through the porous materials. In this paper we consider that the
percolating clusters are filled by gaseous light nanoemitters in the excited
state. A typical incipient percolation cluster has a dendrite shape. From Fig.
1 (a) we observe that such a system consists of two essentially different
areas. In the first region (in the vicinity of the entrance shown by arrow)
there is a strong concentration of percolation clusters, which may lead to
accumulation of the nonlinear field effects and the lasing. In the other part,
the concentration of percolation clusters is small, so the disorder of medium
is lower. We study the emission of electromagnetic energy from a cubical
sample $(x,y,z)\in[0,l_{0}]$ in 3D medium with a percolating disorder, where
the percolating clusters are filled by the light nanoemitters in the excited
state. The output flux of energy can be written as
$I=\oint_{S}(\mathbf{K}\cdot\mathbf{n})dS=I_{x}+I_{y}+I_{z}$, where
$\mathbf{K}$ is the Pointing vector, $\mathbf{n}$ is the normal unit vector to
the surface $S$ of cube, and $I_{x,y,z}$ indicate the fluxes from two faces of
the cube perpendicular to a particular direction. To find the emission from
the system we solve numerically the following equations that couple the
polarization density $\mathbf{P}$, the electric field $\mathbf{E}$, and
occupations of the levels of emitters $N_{i}$ Siegman:1986

$\frac{\partial^{2}\mathbf{P}}{\partial
t^{2}}+\Delta\omega_{a}\frac{\partial\mathbf{P}}{\partial
t}+{\omega_{a}^{2}}\mathbf{P}=\frac{6\pi\varepsilon_{0}c^{3}}{\tau_{21}\omega_{a}^{2}}(N_{1}-N_{2})\mathbf{E,}$
---
$\frac{\partial N_{0,3}}{\partial t}=\mp
A_{r}N_{0}\pm\frac{N_{1,3}}{\tau_{(10),(32)}},$
$\frac{\partial N_{1,2}}{\partial
t}=\frac{N_{2,1}(t)}{\tau_{(21),(32)}}\mp\frac{(\mathbf{j}\cdot\mathbf{E})}{\hbar\omega_{a}}-\frac{N_{1,2}}{\tau_{(10),(3.2)}},$
(1)

here $\Delta\omega_{a}=\tau_{21}^{-1}+2T_{2}^{-1}$, where $T_{2}$ is the mean
time between dephasing events, $\tau_{21}$ is the decay time from the second
atomic level to the first one, and $\omega_{a}$ is the frequency of radiation.
The electric and magnetic fields, $\mathbf{E}$ and $\mathbf{H}$, and the
current $\mathbf{j}=\partial\mathbf{P}/\partial t$ are found from the Maxwell
equations, together with the equations for the densities $N_{i}(\mathbf{r},t)$
of atoms residing in $i-$th level, see Soukoulis:2000  and references therein.
An external source excites emitters from the ground level ($i=0$) to third
level at a certain rate $A_{r}$ , which is proportional to the pumping
intensity in experiments. After a short lifetime $\tau_{32}$, the emitters
transfer nonradiatively to the second level. The second level and the first
level are the upper and the lower lasing levels, respectively. Emitters can
decay from the upper to the lower level by both spontaneous and stimulated
emission, and $(\mathbf{j}\cdot\mathbf{E})/\hbar\omega_{a}$ is the stimulated
radiation rate. Finally, emitters can decay nonradiatively from the first
level back to the ground level. The lifetimes and energies of upper and lower
lasing levels are $\tau_{21}$, $E_{2}$ and $\tau_{10}$, $E_{1}$, respectively.
The individual frequency of radiation of each emitter is then
$\omega_{a}=\left(E_{2}-E_{1}\right)/\hbar$, and $\hbar$ is Planck’s constant.

Numerics. In our calculations we considered the gain medium with parameters
close $GaN$ powder, similar to Ref. expt . The lasing frequency $\omega_{a}$
is $2\pi\times 3\times 10^{13}\,Hz$, the lifetimes are $\tau_{32}=0.3\,ps$,
$\tau_{10}=1.6\,ps$, $\tau_{21}=16.6\,ps$, and the dephasing time is
$T_{2}=0.0218\,ps$. The percolating cluster has been generated inside the cube
of $l_{0}=1\,\mu m$ edge having $L^{3}$ nodes, with $L=100$ (see Fig.1 (a))
that was sufficient to simulate the percolating structure Burlak:2009a . Each
node is supposed to indicate the position of many emitters, with the total
concentration of emitters inside the percolation cluster being
$N=N_{0}+N_{1}+N_{2}+N_{3}=3.3\times 10^{24}\,m^{-3}$. The initial (at $t=0$)
values of densities $N_{0}(0)=0.001\,N$, $N_{1}(0)=0.002\,N$,
$N_{2}(0)=0.002\,N$, and $N_{3}(0)=0.995\,N$ are used. The dielectric
permittivity of a host material is $n=2.2$ that is close to the typical values
for ceramics $Lu_{3}Al_{5}O_{12}$, $SrTiO_{3}$, $ZrO_{2}$, see review
Sanghera:2012 . The result of simulations shown in Fig. 1 (b) obtained for cw
pumping given by $A_{r}=10^{7}\,s^{-1}$. At this pumping the simulations show
the formation a well-defined lasing for $t>t_{s}$, and we refer $t_{s}$ as the
lasing start time. To simulate the noise in our system the initial seed for
the electromagnetic field has been created with random phases at each node. In
what follows it is used dimensionless time $t$ renormalized as $t\rightarrow
tc/l_{0}$,where $c$ is the light velocity in vacuum.

Briefly, the strategy of our simulations consists in the following: (i)
Calculating the geometry of the spanning percolating cluster, (ii) Calculating
the photon field $\mathbf{E}$ generated by emitters incorporated in spanning
cluster with the use of finite-difference time-domain (FDTD) technique
AllenTaflove:2005a , and (iii) Solution of nonlinear dynamic coupled equations
for field, polarization $\mathbf{P}$ and the occupation numbers $N_{i}$ for
all the emitters in 3D system.

Figure 1: (Color online.) (a) Typical spatial structure of the incipient
percolating cluster near the percolation threshold in the cube $l_{0}\times
l_{0}\times l_{0}$. (b) The dynamics of lasing (the critical lasing time is
$t_{c}=5$), where the field fluxes $I_{xyz}$ in the $xyz$ -directions of the
system are displayed; (c) The normalized fields in the central intersection
$xy$ of 3D system for various times closely to the critical time at: (c)
$t=4<t_{c}$, and (d) $t=5=t_{c}$. In such situation only narrow point-like
fields generated by random emitters are observed. For longer time at
$t=6>t_{c}$ (e) the field bunches having smooth localized shape and large
amplitudes arise already beyond the emitter peaks. Hence the lasing in
percolating disordered system will stimulate the localization of emitted
field.

We use here the extended 3D percolating model that allows the empty
percolating voids to have a small random-valued 3D shift from the reference
grid. We use the percolating probability $p=0.4$ that is slightly less of
corresponding threshold $0.415$ for such extended model. Fig.1 (a) shows the
typical spatial structure of the incipient percolating cluster in the cube
$l_{0}\times l_{0}\times l_{0}$ corresponding to the numerical grid with
$L\times L\times L$, $L=100$ nodes. Fig.1 (b) displays the dynamics of optical
lasing generated by exited emitters in percolating medium with intensities
$I_{x,y,z}$ through the sides of 3D sample. The normalized field amplitudes
$E_{x}$ in the central intersection ($x,y=L/2$) of 3D percolating system at
different times nearly of the initial time of laser generation ($t=4,5,6$) are
shown in Fig.1, panels (c), (d), and (e). Fig.1 (c) shows the fields
distribution before the lasing starts. In this case the field amplitudes are
small, one can observe only narrow point-like fields (indicated by arrow)
generated by random emitters in percolating cluster. At larger time $t=5$ (see
Fig.1, panel (d)) the laser generation starts that leads to increase of the
field amplitudes in the emitter positions, however the radiation field still
is small. But for $t=6$ (peak of lasing, see Fig.1, (b), (e)) we observe the
appearance of intensive radiating light beyond the emitter peaks.

Figure 2: (Color online.) The field distribution for longer times. Panels (a)
$t=7$ and (b) $t=8$ show the dynamic states (coupled to emitters) when the
light bunches begin to migrate from the random emitter area. (That corresponds
to lasing evolution in Fig.1 (b) for the interval $t=7,8$ when in the system
after emission the accumulation of energy occurs. Panels (c) $t=9$ and (d)
$t=10$ show that for long time the optical bunches reach the output of system,
and at $t=10$ (d) the localized light is radiated to surrounding space.

Fig.2 shows the developed field distribution for larger times: $t=7,8,9,10$,
when the advanced lasing occurs. From this figure one can determine $t=t_{D}$
as a maximum dwell time such that for $t\leq t_{D}$ the dynamic bunches are
still formed in all the nonlinear and disordered part of system from $x=0$ up
to $x=L/2$ and for $t>t_{D}$ the fiel bunches are concentrated at $x>L/2$.
Figures 2 (a) and (b) show the nonlinear dynamics at $t=7$ and $t=8$ when
well-formed field bunches with high amplitude start to leave the random
emitter area . This corresponds to field evolution in Fig.1 (b) for $t=7,8$
when the system after emission is in the state of the energy accumulation.
Fig.2 (c), (d) show that for longer time the field bunches move to output of
system; at $t=9$ they are closely to the output and first front starts to
leave the system. Moreover, we observe from Fig.2 (d) that at $t=10$ the
bunches have been radiated away the system such that the field tails have
quite small amplitude, therefore the points-like emitter fields (close to the
input) are seen in this amplitude scale. We observe that for the used
parameters the dwell time is $t_{D}\approx 7$.

Figure 3: (Color online.) The isosurface of 3D emitted field shown in Fig. 2
(a) for time $t=7$. We observe that the field has indented shape of light
structures with bunches and worm-holes due to the fractality of radiated
system.

Fig.2 exhibits the fields in central intersection of 3D system. However, since
the clusters have nonuniform fractal structure it is more informative to
explore the shape of fields in total 3D volume. Fig. 3 displays the isosurface
of field in 3D system shown in Fig. 2 (a) for time $t=t_{D}=7$ when the
optical fields have already well-established structure. We observe from Fig. 3
that in general field consists of bunches having different amplitudes and
shapes. The bunches are separated by 3D worm-holes (with small field) having
irregular shapes that emerge due to the non-integer fractal dimension
($D_{H}=2.54$) of percolating clusters.

Localization. In the above the properties of generated field bunches were
analyzed in general. In what follows we study the dynamics and structure of
localized fields confined in the light bunches. A field (quasi-spherical)
bunch will be mentioned as a bounded 3D domain where field Ei in every layer i
is a decreasing function $Ei>Ei+1$ $(i\in 0,1,\ldots p-1$) from center to
periphery that leads to $E_{0}>E_{1}>\ldots>E_{p}>0$. Such domain (bunch) have
two parameters: radius $p$ and the field value in the center $E_{0}=E_{max}$.
[More detailed approaches in such 3D localization problem require too large
amount of computer resources.]

Figure 4: (Color online.) The positions and shapes of some localized field
bunches with highest (normalized) amplitudes in center $E_{max}$ and largest
radius $p$ for $t=t_{D}=7$ (see Fig. 2 (a)) situated in different points of 3D
system: (a) $E_{max}=0.98$ at $x=14,y=18,z=73,p=5$; (b) $E_{max}=0.96$ at
$x=13,y=26,z=18,p=5$; (c) $E_{max}=0.73$ at $x=15,y=41,z=64,p=4$; (d)
$E_{max}=0.74$ at $x=15,y=41,z=64,p=4$; (e) $E_{max}=0.5$ at
$x=73,y=20,z=32,p=5$ , and (f) $E_{max}=0.55$ at $x=18,y=35,z=21,p=5$. In
general case the bunches are situated in random positions due to disordered
radiated emitters and the fractality of percolating clusters.

From Fig. 4 one observes for time $t=t_{D}=7$ the structure of some localized
field with large amplitudes $E_{max}$ and characteristic radius $p$. The
centers of such objects are established in various points of 3D medium. Fig. 4
displays the isosurface of normalized fields for such items. In general the
bunches are situated in variety points of 3D system, see Fig. 3 due to
disordering positions of radiated emitters and the fractal shape of the
percolating clusters. As we observer from Fig. 4 all the bunches have well
established localized shape.

Figure 5: (Color online.) Dependence the number of optical bunches $N$ on the
radius $p$ for various times $t=4,5,6,7,8,9,10$. We observe that $N$ with
large radius $p$ drastically increments after $t>5$ when the lasing starts. At
$t=t_{D}=7$ and $t=8$ it is generated the bunch with maximal characteristic
radius $p=7$.

It is interesting to study the dependence of number of bunches at different
times $t$ as function of radius $p$. Fig.5 shows such dependence for times
$t=4,5,6,7,8,9,10$. We observe from Fig.5 that below the lasing threshold
($t\leq 5$, see Fig. 1 (b)) there are large number of bunches with small
radius $p$. Most of them correspond to narrow point-like fields generated in
vicinity of random light emitters (see Fig. 1 (c)). However, as one can see
from Fig.5, after the lasing starts (since $t>5$) the number of optical
bunches with large radius $p$ drastically increments. Besides, Fig. 2 displays
that such bunches (wth large $p$) are situated mainly beyond the percolating
area. Moreover, as Fig. 2 shows such fields have well-localized shape (Fig.
4), they are not pinned to the static emitters and migrate to the output. That
allows indicating such dynamic bunches as 3D zones of optical Anderson
localization arisen by the active disordered fractal percolating clusters.

Finally it is especially interesting to study the dependence amount of bunches
at different times $t$ as function of radius $p$. Fig.5 shows such dependence
for times $t=4,5,6,7,8,9,10$. We observe from Fig.5 that below the lasing
threshold ($t\leq 5$, see Fig. 1 (b)) there are a lot of field peaks with
small radius $p$. Most of them correspond to narrow point-like fields
generated by random light emitters (see Fig. 1 (c)). However, as one can see
from Fig.5, after the lasing starts (since $t>5$) the number of optical
bunches with large radius $p$ drastically increments. Besides, Fig. 2 displays
that such bunches (with large $p$) are situated mainly beyond the percolating
area. Moreover, as Fig. 2 shows such fields have well-localized shape (Fig.
4), they longer are not pinned to the static emitters and they migrate to the
output of system. That allows indicating such dynamic bunches as 3D zones of
optical Anderson localization arisen by the active fractal percolating
clusters.

Conclusion. We have shown that dynamic 3D optical Anderson localization can
arise in medium with percolating disorder when the percolating clusters are
filled by the light nanoemitters in the excited state. In such materials the
field emitters in clusters create a fractal radiating system, where the field
is not only emitted, but also scattered by the boundaries of clusters. Our 3D
simulations have found that in such nonlinear compound the amount of localized
field bunches drastically increases after the optical lasing starts. At longer
times all the bunches leave nonlinear percolating structure and are radiated
from the medium.

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# DataPop: Knowledge Base Population using Distributed Voice Enabled Devices

Elena Montes, Monique Shotande, Daniel Helm, Christan Grant
{elenamontes, monique.shotande, daniel.helm<EMAIL_ADDRESS>
###### Abstract

Data scientists are constantly creating methods to efficiently and accurately
populate big data sets for use in large-scale applications. Many recent
efforts utilize crowd-sourcing and textual interfaces. In this paper, we
propose a new method of curating data; namely, creating a multi-device Amazon
Alexa Skill in the form of a research trivia game. Users experience a
synchronized gaming experience with other Amazon Echo users, competing against
one another while filling in gaps of a connected knowledge base. This allows
for full exploitation of the speed improvement offered by voice interface
technology in a game-based format.

##  1 Introduction

Knowledge Bases are increasingly becoming a structure for storing structured
information. Researchers have developed methods to gather information from the
various sources to create knowledge bases Banko et al. (2007); Carlson et al.
(2010); Chen et al. (2016). However, these methods require significant post-
processing, including probabilistic evaluation to ensure the gathered facts
are correct Doan et al. (2011); Nakashole and Mitchell (2014); Dong et al.
(2015). Current methods for populating knowledge bases use textual or
graphical user interfaces to enable crowd-sourced input Crescenzi et al.
(2017). This team proposes a new method that utilizes a voice interface and
game format for knowledge base population. By creating an Alexa Skill and
synchronized back end, multiple Amazon Echo voice enabled devices can
simultaneously interact, allowing users to play a research trivia game
together. Given an incomplete knowledge base, natural language questions are
posed to the set of users currently playing a game. Each submitted answer is
collected and the most probable answer is computed using a combination of user
agreement weighted by the accuracy of each user from previous ground truth
questions. In this paper, we will motivate the need for a voice based
knowledge population game (§ 2). We will then, describe the architecture (§ 3)
and game play details (§ 4). We will conclude with a description of the demo
(§ 5) and summary (§ 6).

##  2 Motivation

Traditional methods for knowledge base population rely on textual input.
Whether from mobile or desktop devices, users must type in pieces of
information to be sent in for review. This process is tedious for many users,
particularly when dealing with large data sets. Research from Ruan et al.
(2016) finds that voice input can be up to three times faster than textual
input. New voice interface technologies used in devices, such as the Amazon
Echo 111https://www.amazon.com/echo or Google Home 222https://home.google.com,
offer an opportunity to significantly increase input speed of knowledge base
population.

Many large knowledege bases are constructed by bootstrapping existing large
knowledge bases Rodríguez et al. (2016) — Freebase
333https://developers.google.com/freebase/, NELL
444http://rtw.ml.cmu.edu/rtw/, OpenIE
555https://nlp.stanford.edu/software/openie.html, ProBase
666https://www.microsoft.com/en-us/research/project/probase/, ProbKB
777https://dsr.cise.ufl.edu/projects/probkb-web-scale-probabilistic-knowledge-
base/, and YAGO 888http://www.mpi-inf.mpg.de/departments/databases-and-
information-systems/research/yago-naga/yago/ are some of the most popular
bootstrapped knowledge bases. These knowledge bases are currently popular, but
many industries use the relatively smaller, topical knowledge bases such as
those generated from smaller data sets or crowdsourced users imput Steiner and
Mirea (2012); Wang et al. (2013). In both large and small knowledge bases, the
fusion of facts can create conflicting sets of information and the
subsequently uncertain or outdated information Dong et al. (2014); Grant and
Wang (2015). This motivated the need and usefulness for storing the
probabilities or likelihoods associated with each fact.

Many knowledge bases use paid services like Amazon Mechanical Turk for data
population. When large data sets need to be populated for large-scale
applications, especially within the realm of university work or teams with a
small budget, this route is simply not fiscally feasible. With the aid of a
free and open source game, for example, these teams can encourage users to
assist in knowledge base population without a monetary incentive. The solution
to the latter is to seek curating from a base of users, without a monetary
incentive. This form of basic research data entry with no discernible reward,
risks user boredom or fatigue. Results may be slow or inaccurate.
Gamificiation, when implemented properly, has been shown to increase user
engagement and create feasible reward systems Hamari (2017).

##  3 Architecture

This application relies on each user possessing on Amazon smart speaker device
— this may includes an Amazon Echo, Amazon Echo Dot, or an Amazon Tap. Each
device is web enabled and contains speaker and a microphone. These devices act
as the user interface for the system. Each user must downlaod and install a
DataPop Amazon Alexa Skill. The skill allows the user to interact with the
device, voice questions, and collects responses. A program running on our
server handles synchronization to group disparate devices/users into games and
keeps track of relevant information, such as start/end timestamps and
cumulative scores. In addition, this piece keeps track of all existing users,
attaching to their unique identifiers and key information such as accuracy
score, the number of answered questions, and current games, if any.

To generate questions, DataPop has an interface to a knowledge base. The
program can and searches for missing or low confidence entries in the
knowledge base. The system can then generate questions from a structured
format called Intents, and are sent to each Alexa system playing in the game.
The system then expects to receive answers in a particular format by specify
the Slots Type. We discuss this approach in detail in § 4.

###  3.1 Hardware

The setup for the game interface requires multiple Amazon Echo devices,
alongside a server (optional - AWS or a similar service may be used), which
runs the Alexa Skill and the backend synchronized gameplay application.

Figure 1: Hardware Setup and Attributes

####  3.1.1 Setup Interaction

Self-motivated users are the most engaged and the best for populating large
data sets. We adapt select questions for users that will most interest them
Asking relevant questions keeps users engaged over time. Additionally,
Tailoring the types of questions a user will answer to those that are relevant
to their interests and preferences will increase the likelihood of accurate
responses and the speed at which responses are obtained.

During the Alexa Skill setup interaction (which only happens once), the user
first specifies their interests upon “signing up” for this Skill. The person
will then answer a series of questions to gauge his/her areas of interests. To
predict user interests, we first train an five-layer neural network to accept
five to ten phrases words, find the average location of those words in a word
embedding. The training process minimizes the distance between the metric
spance and the labeled category. When a set of phrases of user interest are
submitted, the are fit to a set of categories, e.g., Artificial intelligence,
Databases, Information Retrieval. The categories represent a category name or
predetermined cluster of a partition of the knowledge base.

####  3.1.2 Further Skill Interaction

Once setup is completed, the skill must be invoked each time a user wishes to
enter a game. The phrase “Alexa, ask DataPop to let me join a game” or similar
phrases will begin this process.

###  3.2 Knowledge Base Preparation

Before the game can be played, the knowledge base chosen for gameplay must be
prepared. The knowledge base is partitioned into categories either from
clustering or using an attribute of each item in a knowledge base. It is
instructive to picture the knowledge base as having both a row and column
type. Columns in the knowledge base correspond to a set of related information
and rows correspond to an entity. For a knowledge base about people, columns
may refer to names, dates of births, and telephone numbers.

In order to efficiently fill in the data set, some priorities must be set
corresponding to to complexity of obtaining an answer in the data set. For
example, in the person data set, dates of birth are often easier to locate
than the elementary school a person attended, and thus, the former would
receive a lower difficulty label. All the columns in the knowledge base will
be ranked numerically, with zero being the most difficult and 1 being the
least difficult.

Question templates, or Intents, are manually created for two or more column
types. For example, a question template, or intent may be:

What is the {Column Name} of {Entity Name}?

or

When did {Entity Name} join the {Column Name} {Column Value}?.

Entity Name can be a pseudo key or name in the knowledge base such as Jeremy
Gibson or Javier Lopez. The Column Name can be Slot types such as a date (in
Alexa nomenclature, Amazon.DATE or AMAZON.EducationalOrganization) and the
Column Value can be a value such as 2013 or CMU as seen in Figure 2.

##  4 Gameplay

Many aspects of the program are handled during live gameplay.

###  4.1 Game Skill Interface

Questions posed to users will be formed based on data fields that are either
missing or have insufficient confidence. Users will have the option to either
skip or answer any questions posed to them. Once a user answers a question, it
will be recorded with a level of confidence proportional to the user’s
accuracy. Because of a limitation in the Alexa system, the user has no more
than 300 seconds to respond to a question; if no response is given within this
time, the query is marked as unanswered and gameplay continues. The user can
always ask Alexa for their current score or for more time.

###  4.2 Query Creation

Queries are automatically generated based on a filtered-down collection of
gaps detected in the knowledge base. From these groupings, the designers of
the knowledge base can hard-code query templates. As the knowledge base grows,
generating new templates is simple. Each question has an associated response
type as an Amazon Slot. Associating a response with an type improves the
accuracy of the voiced answer content.

The filtering of these gaps to choose a particular query to generate happens
in stages. First, a collection of rows with missing information is chosen
which corresponds to the topic of the current game. Then, if there is more
than one row returned from this operation, the results are further filtered by
a priority ranking. Each column in the data set is ranked based on its
importance. Finally, only rows where the gap in the data occurs under the most
important column, are chosen. If there are still multiple rows to choose from,
we make a random choice and generate a query on that gap in the knowledge
base.

Figure 2: Example Query Generation

###  4.3 Synchronization

Users can compete with their friends or random players in different
categories, by answering the same set of questions. Each user has 300 seconds
to answer (or skip) each question. Players will be informed of their
opponent’s answers, and overall performance, after both users have responded.

####  4.3.1 Player Ranking

Fueling a bit of competition between players will encourage continuing
gameplay and long-term interest in the Skill. Using the method outlined in S̃
4.4, users’ accuracy scores are continually updated to help with the ranking
process at the end of a game. Higher-ranked players will receive more rewards
in the form of points, and the ’winner’ of the round (with the most likely
correct answer), also receives more points.

####  4.3.2 Badges

As players accumulate points over time and work on improving their accuracy
scores, they will receive audio badges that are stored on their profile. Part
of the Skill’s frontend allows players to check on their badge collection at
any time and hear Alexa read off a listing of badges. This contributes to the
gaming experience and works as an incentive for players to continue curating
the knowledge base.

###  4.4 Accuracy Scoring

Two problems arise in a gamified knowledge base population: resolving
conflicts and encouraging continued user participation. Giving users an
accuracy score will help to resolve both issues; the former by contributing to
a probabilistic determination of the most likely correct answer, and the
latter by using accuracy scores to dole out badges and create some competition
between game players.

Throughout gameplay, we mix in a number of predetermined queries with known —
and correct responses. A user’s responses to these queries contribute to their
overall accuracy score. Correct responses contribute positively, with higher
difficulties gaining more points, while incorrect responses contribute zero
points.

If the answer is correct, this simple formula is used to accumulate user
points:

$\text{user}_{\text{acc}}=\text{user}_{\text{accuracy}}+\text{column}_{\text{difficulty}}$

Figure 3: User Accuracy Scoring

###  4.5 Confidence Scoring

In order to account for the users providing multiple distinct results as
potentially correct answers to a query, we devise a system of confidence
ranking that considers the users’ accuracy scores and how often an answer has
been seen.

The average accuracy of the users, at the time of response to the question, is
used to rank each answer for any given question:

$c_{i}=\frac{\sum_{j=1}^{n}ua_{ij}}{\sum_{k=1}^{m}ua_{k}}$

Figure 4: Confidence Scoring

Where $c_{i}$ is the confidence for the ith answer; $ua_{ij}$ is the accuracy
for the jth user that provided the ith answer; $ua_{k}$ is the accuracy for
the user kth user that answered this question; n is the number of users that
provided answer i; and m is the number of users that provided answers to the
question.

For example, suppose two separate answers are provided for “What is the rank
of Jeremy Gibson?” Two users with accuracy scores of .6 and .8 provided the
answer associate, and three users with accuracy scores .5, .7, and .9 provided
the answer full. The confidence allotted to the associate answer will be
$(.6+.8)/(.5+.6+.7+.8+.9)=.4$; the confidence allotted to the full answer will
be $(.5+.7+.9)/(.5+.6+.7+.8+.9)=.6$.

##  5 Demo Scenario

Two or three devices will be set up, at a distance from one another -
preferably, one will be in another room while videoconferencing in. There will
be one user paired to each device, and these users will go through the setup
process to establish their game profiles.

A game will begin and the audience will be able to see how each user is asked
the same questions and alerted to their scores at the end. If all users are
not perfectly synchronized during gameplay - which is a scenario to be
expected - they will receive an update notification when the game has
concluded for all players. At this time, overall prizes are announced for the
winner of the round. A laptop will be used to allow attendees to modify and
explore the knowledge base.

Additional features such as hearing a badge summary and help summary will also
be demonstrated.

##  6 Summary

A gamified Alexa skill may be used in other areas where humans bring a unique
ability to quickly interpret natural language and recognize patterns, such as
in text summarization.

One weakness of this system is the limited time available for user responses.
With a time constraint on answering the query Alexa gives them, the user may
not be able to deliver accurate results, especially if they feel under stress.

Our future plans are to expand this gamified knowledge base population method
to a more generalized model that can be easily adapted to fit any existing
knowledge base.

## References

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