# VIRTUAL CONTINUITY OF MEASURABLE FUNCTIONS OF SEVERAL VARIABLES AND
EMBEDDINGS THEOREMS

A. M. Vershik, P. B. Zatitskiy, F. V. Petrov

###### Abstract

Classical Luzin’s theorem states that the measurable function of one variable
is “almost” continuous. This is not so anymore for functions of several
variables. The search of right analogue of the Luzin theorem leads to a notion
of virtually continuous functions of several variables. This probably new
notion appears implicitly in the statements like embeddings theorems and
traces theorems for Sobolev spaces. In fact, it reveals their nature as
theorems about virtual continuity. This notion is especially useful for the
study and classification of measurable functions, as well as in some questions
on dynamical systems, polymorphisms and bistochastic measures. In this work we
recall necessary definitions and properties of admissible metrics, define
virtual continuity, describe some of applications. Detailed analysis is to be
presented in another paper.

†† St. Petersburg Department of V.A.Steklov Institute of Mathematics RAS, St.
Petersburg State University. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>The work is supported by RFBR grant 11-01-00677-a,
President of Russia grant MK-6133.2013.1, by Chebyshev Laboratory in SPbSU,
Russian Government grant 11.G34.31.0026.

##  1 Introduction. Admissible metrics, Luzin’s theorem.

###  1.1 Admissible metrics

We consider the Lebesgue-Rokhlin standard continuous (atomless) probabilistic
measure space, isomorphic to the unit segment $[0,1]$ with Lebesgue measure.
The first author proposes [4, 8, 11] to consider on the fixed standard space
$(X,{\mathfrak{A}},\mu)$ different (admissible) metrics, on the contrast to
usual approach, when metric space is fixed and different Borel measures are
considered. Such approach is useful and necessary in ergodic theory and other
situations. Agreement of the metric and measure structures leads to the notion
of admissible metric triple:

######  Definition 1.

A metric or semimetric $\rho$ on the space $X$ is called _admissible_ if it is
measurable, regarded as a function of two variables, on the Lebesgue space
$(X\times X,\mu\times\mu)$ and there exists a subset $X_{0}\subset X$ of full
measure such that the semimetric space $(X_{0},\rho)$ is separable.

The standard probabilistic space $(X,\mu)$ with admissible (semi-)metric
$\rho$ is called admissible metric triple or just admissible triple
$(X,\mu,\rho)$.

Properties of admissible metrics are studied in details by the authors in
[12], [17]. In particular, these works contain several equivalent definitions
of admissible triples.

Standartness of the space allows to get the following

######  Proposition 1.

Let $\rho$ be the admissible metric on $(X,{\mathfrak{A}},\mu)$. Then
completed Borel sigma-algebra ${\mathfrak{B}}={\mathfrak{B}}(X,\rho)$ is a
subalgebra of ${\mathfrak{A}}$. The measure $\mu$ is inner regular with
respect to metric $\rho$, i.e. for any $A\in{\mathfrak{A}}$ we have

$\mu(A)=\sup\\{\mu(K)\colon K\subset A,K\mbox{ is compact in metric }\rho\\}.$

So, for any admissible metric $\rho$ the measure $\mu$ is Radon measure in the
metric space $(X,\rho)$.

M. Gromov in the book [3] suggests to consider arbitrary metric triples
$(X,\mu,\rho)$, which he calls $mm$-spaces. Also, Gromov asks the question
about their classification, having in mind classical situations (Riemannian
manifolds and so on). It is natural to consider admissible triples in this
framework. Define equivalence of admissible triples up to measure-preserving
isometries: $(X,\mu,\rho)\sim(X^{\prime},\mu^{\prime},\rho^{\prime})$, if

$\exists T:X\rightarrow X^{\prime};\quad
T\mu=\mu^{\prime};\quad\rho^{\prime}(Tx,Ty)=\rho(x,y).$

Here is the main result on this equivalence:

######  Theorem 1.

(Gromov [3]; Vershik [4])

Consider the map $F_{\rho}:X^{\infty}\times X^{\infty}\rightarrow
M_{\infty}(\mathbb{R}):$

$\quad
F_{\rho}(\\{x_{i},y_{j}\\}_{(i,j)\in\mathbb{N}\times\mathbb{N}})=\\{\rho(x_{i},y_{j})\\}_{(i,j)\in\mathbb{N}\times\mathbb{N}},$

and equip infinite product $X^{\infty}\times X^{\infty}$ by the product-
measure $\mu^{\infty}\times\mu^{\infty}$. Let $D_{\rho}$ denote the measure on
the space of matrices (i.e. random matrix of distances), which is the
$F_{\rho}$-image of the measure $\mu^{\infty}\times\mu^{\infty}$. Call it
MATRIX DISTRIBUTION of the metric $\rho$. It is a complete invariant of above
equivalence of admissible metrics.

In other words,

$(X,\mu,\rho)\sim(X^{\prime},\mu^{\prime},\rho^{\prime})\Leftrightarrow
D_{\rho}=D_{\rho^{\prime}}.$

In [5] this result is generalized to the so called _pure_ measurable functions
of several variables.

The following lemma is useful in the theory of admissible metrics:

######  Lemma 2.

Let $\rho_{1}$, $\rho_{2}$ be admissible semimetrics on the standard space
$(X,\mu)$, and suppose that $\rho_{1}$ is metric. Then for any $\varepsilon>0$
there exists measurable subset $K\subset X$ such that $\mu(K)>1-\varepsilon$
and semimetric $\rho_{2}$ (as a function of two variables) is continuous on
$K\times K$ with respect to metric $\rho_{1}$.

We may choose $K$ as a compact subset with respect to admissible metric
$\rho=\rho_{1}+\rho_{2}$, if $\mu(K)>1-\varepsilon$.

Lemma immediately implies the

######  Corollary 3.

Let $\rho_{1}$ and $\rho_{2}$ be two admissible metrics on the standard space
$(X,\mu)$. Then for any $\varepsilon>0$ there exists $K\subset X$ such that
$\mu(K)>1-\varepsilon$ and topologies defined by metrics $\rho_{1}$ and
$\rho_{2}$ on $K$ coincide.

###  1.2 Luzin’s theorem on measurable functions of one variable

Furthermore we consider (measurable) real-valued functions, though most of our
results remain true for maps into standard Borel space, in particular into
Polish spaces. Egorov’s and Luzin’s classical theorems on measurable functions
of one variable are well-known. The generalized Luzin’s theorem for arbitrary
admissible triple follows from above results:

######  Corollary 4 (Luzin’s theorem).

Let $\rho$ be an admissible metric on the standard space $(X,\mu)$, let $f$ be
a measurable map from $X$ into Polish space $(M,d)$. Then for any
$\varepsilon>0$ there exists a measurable subset $K\subset X$ such that
$\mu(K)>1-\varepsilon$ and $f$ is continuous on $K$ with respect to metric
$\rho$.

###### Proof.

Set $\rho_{1}(x,y)=\rho(x,y)+d(f(x),f(y))$. Then $\rho_{1}$ is a trivial
example of an admissible metric, with respect to which $f$ is continuous. By 3
there exist a subset $K$ having measure $\mu(K)>1-\varepsilon$, on which this
continuity implies continuity with respect to $\rho$. ∎

But this fact does not hold true for functions of several variables.

##  2 Virtual continuity

###  2.1 Definitions and first examples

Let $f(\cdot,\cdot)$ be a measurable function of two variables. Then Luzin’s
theorem analogue (continuity on the product $X^{\prime}\times Y^{\prime}$ of
sets of measure $>1-\varepsilon$ with respect to given metric
$\rho[(x_{1},y_{1}),(x_{2},y_{2})]=\rho_{X}(x_{1},x_{2})+\rho_{Y}(y_{1},y_{2})$)
is not in general true. This leads to the following key notion of this work.
(Sum of metrics may be replaced to maximum or other metric defining the
topology of direct product. To stress this we denote generic metric with such
topology by $\rho_{X}\times\rho_{Y}$).

######  Definition 2.

Measurable function $f(\cdot,\cdot)$ on the product $(X,\mu)\times(Y,\nu)$ of
standard spaces is called _virtually continuous_ , if for any $\varepsilon>0$
there exist sets $X^{\prime}\subset X,Y^{\prime}\subset Y$, each of which
having measure $1-\varepsilon$, and admissible semimetrics $\rho_{X},\rho_{Y}$
on $X^{\prime},Y^{\prime}$ respectively such that function $f$ is continuous
on $(X^{\prime}\times Y^{\prime},\rho_{X}\times\rho_{Y})$. virtual functions
of several variables are defined in the same way.

It is essential that admissible metric with respect to which function becomes
continuous is not arbitrary, but respects the structure of direct product (in
more general setting, it respects selected subalgebras, see further). It is
easy to verify that there does not exist universal metric of such type (i.e.
such a metric that virtual continuity implies continuity in this metric). It
explains the non-trivial properties of defined notion.

It is clear that any admissible metric (considered as a function of two
variables) is virtually continuous. So is any function, which is continuous
with the respect to product of admissible metrics. Degenerated functions (or
“finite rank functions”) $f(x,y)=\sum_{i=1}^{n}\varphi_{i}(x)\psi_{i}(y)$,
where $\phi_{i}(\cdot),\psi_{i}(\cdot)$, $i=1,\dots n$ are arbitrary
measurable functions, are also virtually continuous. For the proof just use
Luzin’s theorem for all functions $\varphi_{i}(\cdot)$, $i=1\dots n$, and
$\psi_{i}(\cdot)$, $i=1\dots n$.

less trivial examples of virtually continuous functions are given by functions
from some Sobolev spaces and kernels of trace class operators. For virtually
continuous functions there exist well-define restrictions on some subsets of
zero measure — concretely, onto supporters of (quasi)bistochastic measures,
see next paragraph.

An easy example of not virtually continuous measurable function on $[0,1]^{2}$
is provided by the characteristic function of the triangle $\\{x\geq y\\}$. In
general, for functions on the square of a compact group depending of the ratio
of variables the criterion of virtual continuity is simple:

######  Proposition 5.

Let $G$ be a metrizable compact group, $f$ be a Haar measurable function on
$G$. Then the function $F(x,y):=f(xy^{-1})$ on $G\times G$ is virtually
continuous if and only if $f$ is equivalent to a continuous function.

Stress once more that the definition of virtual continuity is not topological,
but measure-theoretical in nature. It applies to the choice of various metrics
on the measure space. So, the direct sense of the proposition 5 is that group
structure and measure-theoretical structure allow to reconstruct topology.

###  2.2 Bistochastic measures and polymorphisms

From the measure-theoretical point of view a function of $k$ variables on the
product of standard continuous spaces is nothing but the function on the
standard continuous space (due to isomorphism of all such spaces). In order to
deal with it as a function of $k$ variables, we have to introduce another
category, then just measurable spaces.

namely, consider the following structure: the measure space $({\cal
X},{\mathfrak{A}},m)$, with $k$ selected sigma-subalgebras
${\mathfrak{A}_{1}},\dots,{\mathfrak{A}_{k}}$ in $\mathfrak{A}$. It is natural
to suppose that those subalgebras generate the whole sigma-algebra
$\mathfrak{A}$.

The connection with general viewpoint is the following: in the space ${\cal
X}=\prod_{i=1}^{k}(X_{i},{\mathfrak{A}_{i}},\mu_{i})$, $m=\prod\mu_{i}$,
identify algebras ${\mathfrak{A}_{i}}$ with subalgebras of
${\mathfrak{A}}=\prod{\mathfrak{A}}_{i}$ by multiplying to trivial subalgebras
on other multiples. In other words, function $f(x_{1},\dots,x_{k})$ on ${\cal
X}$ is ${\mathfrak{A}}_{i}$-measurable iff $f$ depends only on $i$-th variable
$x_{i}$ ($i=1,\dots,k$).

######  Definition 3.

Measurable function on the standard space ${\cal X}$ with $k$ selected
subalgebras, which generate the whole sigma-algebra, is called general
measurable function of $k$ variables.

Consider some measure $\lambda$ on the sigma-algebra $\mathfrak{A}$. It may be
restricted onto sigma-subalgebras ${\mathfrak{A}}_{i}$, $i=1,\dots,k$. Let’s
consider such measures $\lambda$ for which those restrictions are absolutely
continuous with respect to the measure $m$ (restricted onto
${\mathfrak{A}}_{i}$). If restrictions of $\lambda$ onto ${\mathfrak{A}}_{i}$
coincide with $m$, $i=1\dots k$, such a measure $\lambda$ is called
multistochastic with respect to given subalgebras (bistochastic for $k=2$); if
restrictions are just equivalent to $m$ for $i=1,\dots,k$, we call $\lambda$
almost multistochastic. Finally, if $\lambda(U)\leq m(U)$ for any
$U\in{\mathfrak{A}}_{i},i=1,\dots,k$, we call $\lambda$ submultistochastic.

Of course, bistochastic measure on $X\times Y$ may be singular with respect to
the product measure. For instance, in the case of direct product of segments
$(X,\mu)=(Y,\nu)=[0,1]$ there is a bistochastic measure $\lambda$ on diagonal
$\\{x=y\\}$ (with density $d\mu(x)$).

Furthermore we suppose for simplicity that $k=2$, i.e. consider functions of
two variables. But there is no serious difference for $k>2$. We consider not
only independent variables, most of the notions may be defined for general
pair of sigma-algebras. But even the case of independent variables is often
useful to treat as a general case.

Bistochastic measure on the direct product of spaces define the so called
_polymorphism_ of the space $(X,\mu)$ into $(Y,\nu)$ (see [9]), i.e.
“multivalued mapping” with invariant measure. The case of identified variables
$(X,{\mathfrak{A}},\mu)=(Y,{\mathfrak{B}},\nu)$ is of special interest:
polymorphism in this case generalizes the concept of automorphism of measure
space. Almost bistochastic measures defines a polymorphism with quasi
invariant measure. Bistochastic or almost bistochastic measure $\lambda$
defines also a bilinear (in general case $k$-linear) form
$(f(x),g(y))\rightarrow\int f(x)g(y)d\lambda(x,y)$, corresponding to the so
called Markovian, resp. quasi Markovian operator in corresponding functional
spaces. Note that this operator $U_{\lambda}$ is a contraction, i.e. has norm
at most 1, which preserves the cone of non-negative functions. In the case of
bistochastic measure this operator (as well as adjoint operator) preserve
constants: $U_{\lambda}1=1$.

See [9, 10, 7] about many connections of polymorphisms (Markovian operators,
joinings, couplings, correspondences, Young measures, bibundles etc).
Bistochastic measures play a key role in the intensively developing theory of
continuous graphs [15].

###  2.3 Further properties of virtually continuous functions

First of all, virtually continuous functions enjoy the properties a priori
stronger than required in the definition. On the one side, we may require for
sets $X^{\prime},Y^{\prime}$ from the definition to have full measure:

######  Theorem 2.

Let function $f(\cdot,\cdot)$ be virtually continuous. Then there exist sets
$X^{\prime}\subset X$, $Y^{\prime}\subset Y$ of full measure and admissible
semimetrics $\rho_{X},\rho_{Y}$ on $X^{\prime},Y^{\prime}$ respectively such
that $f$ is continuous on $(X^{\prime}\times
Y^{\prime},\rho_{X}\times\rho_{Y})$.

On the other hand, we may fix arbitrary admissible metrics:

######  Theorem 3.

Let function $f(\cdot,\cdot)$ be virtually continuous. Then for any admissible
semimetrics $\rho_{X},\rho_{Y}$ on $X^{\prime},Y^{\prime}$ and for any
$\varepsilon>0$ there exist sets $X^{\prime}\subset X,Y^{\prime}\subset Y$,
each of which having measure $1-\varepsilon$, such that function $f$ is
continuous on $(X^{\prime}\times Y^{\prime},\rho_{X}\times\rho_{Y})$.

A function of two variables on $X\times Y$ may be treated as a map from $X$
into the space of functions on $Y$ (i.e. $f(x,y)\equiv f_{x}(y)$). In [16]
such a viewpoint is used for classification problem. Virtual continuity is
described in those terms by the following equivalent definition:

######  Theorem 4.

Virtual continuity of the function $f(\cdot,\cdot)$ is equivalent to the
following property of a function: for any $\varepsilon>0$ there exist sets
$X^{\prime}\subset X$, $Y^{\prime}\subset Y$ having measures not less then
$1-\varepsilon$, such that the set of functions $f_{x}(\cdot)$ on $Y^{\prime}$
(variable $x$ runs over $X^{\prime}$) form a totally bounded (precompact)
family in $L_{\infty}(Y^{\prime})$.

This theorem-definition has an important corollary: continuity in one variable
implies virtual continuity.

######  Lemma 6.

Let $(X,\mu)$, $(Y,\nu)$ be standard continuous probabilistic spaces,
$\rho_{Y}$ be an admissible metric on the set $Y^{\prime}$ of full measure.
Let measurable function $f:X\times Y\to\mathbb{R}$ be so that functions
$f(x,\cdot)$ are continuous on $(Y^{\prime},\rho_{Y})$ for $\mu$-almost all
$x\in X$. Then $f$ is virtually continuous.

Theorem 2 immediately implies the converse: for any virtually continuous
function $f$ such a metric $\rho_{Y}$ exists. So, the statement of Lemma 6
(“continuity by appropriate metric on $y$ for almost all fixed $x$”) is
equivalent to virtual continuity.

It’s remarkable that the spaces $X$ and $Y$ (i.e. arguments of the function)
play different roles in this definition. However, a posteriori the property
appears to be symmetric under the change of order of variables. This is
another demonstration of the non-triviality of the virtual continuity concept.

As we have seen, measurable functions $f(\cdot,\cdot)$ are classified by
matrix distributions, i.e. by measures on the space of matrices
$(a_{ij})_{i,j=1}^{\infty}$, induced by the map
$f\rightarrow(a_{i,j}=f(x_{i},y_{j}))$ (points $x_{i}$ in $X$ and $y_{i}$ in
$Y$, $i=1,2,\dots$ are chosen independently). Virtual continuity also may be
characterized on this manner:

######  Theorem 5.

Let $x_{1},x_{2},\dots$ (resp. $y_{1},y_{2},\dots$) be independent random
points in $X$ (resp. in $Y$). Virtual continuity of the measurable function
$f(x,y)$ is equivalent to the following condition: for any $\varepsilon>0$
there exist positive integer $N$ such that the following event has probability
1:

there exist such a partitions of naturals
$\\{1,2,\dots,\\}=\sqcup_{i=0}^{N}A_{i}=\sqcup_{i=0}^{N}B_{i}$ that upper
density of the set $A_{0}\cup B_{0}$ is less than $\varepsilon$ (i.e.
$\limsup|(A_{0}\cup B_{0})\cap[1,n]|/n<\varepsilon$) and
$|f(x_{s},y_{t})-f(x_{r},y_{p})|<\varepsilon$ for all $i,j>0$, $s,r\in A_{i}$,
$p,t\in B_{j}$.

Aforementioned characteristics of the virtual continuity allow to deduce that
virtual continuous functions form a nowhere dense set in the space of all
measurable functions (with measure convergence topology).

###  2.4 Thickness

Consider the space $X\times Y$ with product measure $\mu\times\nu$. Choose two
subalgebras in its sigma-algebra, defined by projections onto $X$ and $Y$. For
measurable set $Z\subset X\times Y$ define its _thickness_
$\operatorname{th}(Z)$ as the infimum of the value $\mu(X_{1})+\nu(Y_{1})$
taken by all pairs of measurable sets $X_{1}\subset X,Y_{1}\subset Y$ such
that

$\mu\times\nu(Z\setminus(X_{1}\times Y\cup X\times Y_{1}))=0.$ (1)

Sets of the form $X_{1}\times Y$, $X\times Y_{1}$ are exactly sets from our
sigma-subalgebras. It allows to define generalized thickness for other
selected subalgebras in the standard space.

The following properties of thickness are immediate:

  * •

thickness of a set does not exceed 1 and equals 0 for and only for sets of
measure 0;

  * •

thickness of a subset does not exceed a thickness of a set;

  * •

thickness of a set does not exceed its measure;

  * •

thickness of a finite or countable union of sets does not exceed sum of
thicknesses.

The following lemma is slightly less obvious.

######  Lemma 1.

If $\operatorname{th}(Z)=0$, then $X_{1},Y_{1}$ of zero measure may be chosen
in the definition 1.

By “arbitrarily thin set” we mean “the set of arbitrarily small thickness”. It
allows to reformulate lemma 1 as follows: if the set may covered may
arbitrarily thin set, then its complement contains mod 0 the product of full
measure sets.

the following equivalent definition of thickness is in some situations more
appropriate for using:

######  Lemma 2.

For any set $Z$ consider pairs of measurable functions $f:X\to[0,1]$,
$g:Y\to[0,1]$, such that $f(x)+g(y)\geq\chi_{Z}(x,y)$ for
$\mu\times\nu$-almost all pairs $(x,y)$. Then thickness of $Z$ is infimum of
the sum of integrals $\int_{X}fd\mu$ and $\int_{Y}gd\nu$.

Applying this lemma and choosing weakly convergent subsequence the lower
semicontinuity of thickness may be proved:

######  Lemma 3.

Let $\\{Z_{n}\\}$ -be increasing sequence of measurable sets,
$Z=\cup_{n}Z_{n}$. Then $\operatorname{th}(Z)=\lim\operatorname{th}(Z_{n})$.

Note that there is no upper semicontinuity: all sets
$\\{(x,y):0<|x-y|<1/n\\}\subset[0,1]^{2}$ have thickness 1, while their
intersection is empty.

Define the convergence of functions “in thickness” analogously to convergence
in measure. This is convergence in the metrizable topology defined by the
following distance:

######  Definition 4.

Define the distance $\tau(f(x,y),g(x,y))$ between arbitrary measurable
functions of two variables as infimum of such $\varepsilon>0$ that
$\operatorname{th}\\{(x,y):|f(x,y)-g(x,y)|>\varepsilon\\}\leq\varepsilon$.

Convergence in this $\tau$-metric implies convergence in measure (but not vice
versa).

Let $\xi_{X}:X=\sqcup_{i=1}^{n}X_{i}$, $\xi_{Y}:Y=\sqcup_{i=1}^{m}Y_{i}$ be
finite partitions of the spaces $X,Y$ respectively onto measurable subsets of
positive measure. Functions which are constant mod $0$ on each product
$X_{i}\times Y_{j}$ are called _step functions_. Finite linear combinations
$\sum_{i=1}^{N}a_{i}(x)b_{i}(y)$ are called _functions of finite rank_. The
set of measurable functions is complete in $\tau$-metric.

The following theorem connects finite rank functions and virtual continuity.

######  Theorem 6.

The $\tau$-closure of the set of step functions (or the set of finite rank
functions) is exactly the set of virtually continuous functions.

This definition of virtual continuity is even more explicitly measure-
theoretical, it does not appeal to metrics at all. Also, it has clear
generalisation to arbitrary pair of sigma-subalgebras. See [13] on close
concepts.

###  2.5 Norm in the space of virtually continuous functions

Defined convergence in $\tau$-metric is a virtual continuity analogue of the
convergence in measure. Known Banach spaces of measurable functions also have
their analogues.

A measurable function $h(\cdot,\cdot)$ on the space $(X\times Y,\mu\times\nu)$
is called subbistochastic, if the measure with $\mu\times\nu$-density
$|h(\cdot,\cdot)|$ is subbistochastic. Denote by $\mathcal{S}$ the set of
subbistochastic functions.

Define a finite or infinite norm of a measurable function $f(\cdot,\cdot)$ as

$\|f\|:=\inf\\{\int_{X}|a(x)|d\mu(x)+\int_{Y}|b(y)|d\nu(y):|f(x,y)|\leq
a(x)+b(y)\text{ a.e.}\\}$

Next theorem is an analogue of known L. V. Kantorovich’s duality theorem [1]
in the mass transportation problem (concretely, of duality between measures
space with Kantorovich distance and and the space of Lipschitz functions, see
also [18]).

######  Theorem 7.

$\|f\|=\sup\\{\int_{X\times Y}|f(x,y)|h(x,y)dxdy:h\in\mathcal{S}\\}.$

Coincidence of infimum and supremum is a duality statement in infinite-
dimensional linear programming. But in our case the proof requires more
delicate arguments than the Monge-Kantorovich problem. The reason is that we
consider $L^{1}$-type space, in which the cone of non-negative functions has
empty interior (on the contrast to the space of continuous functions, used in
the transport problem). This does not allow to apply directly standard
separability theorems.

######  Theorem 8.

The closure of step functions in above norm consists exactly of all virtually
continuous functions having finite norm (in particular, each bounded virtually
continuous function belongs to this closure).

Denote this space by $VC^{1}$. It is an analogue of the space $L^{1}$ for
virtually continuous functions and is a pre-dual for the space of
polymorphisms with bounded densities of projections.

######  Theorem 9.

The space dual to $VC^{1}$ is a space of signed measures $\eta$ on $X\times Y$
with finite norm

$\|\eta\|_{\rm{me}}=\max\\{\Big{|}\Big{|}\frac{\partial
P^{x}_{*}|\eta|}{\partial\mu}\Big{|}\Big{|}_{L^{\infty}(X,\mu)},\Big{|}\Big{|}\frac{\partial
P^{y}_{*}|\eta|}{\partial\nu}\Big{|}\Big{|}_{L^{\infty}(Y,\nu)}\\},$

where $P^{x}$ and $P^{y}$ are projections onto $X$ and $Y$ respectively and
$|\eta|$ is a full variation of a signed measure $\eta$.

######  Corollary 1.

Virtually continuous functions from $VC^{1}$ (in particular, bounded virtually
continuous functions) have a well defined integral not only over sets of
positive measure (as all summable functions do have), but also also over
bistochastic (singular) measures: for instance, over Lebesgue measure on
diagonal $\\{x=y\\}\subset[0,1]^{2}$, or over measure concentrated on a graph
of a map with quasiinvariant measure. To summarize, virtually continuous
functions have traces (restrictions) on diagonal and other subsets in the
sense of Sobolev trace theorems.

##  3 Applications: embeddings theorems, trace theorems and restrictions of
metrics

Here we mention some applications of virtual continuity.

###  3.1 Sobolev spaces and trace theorems

######  Theorem 10.

Let $\Omega_{1},\Omega_{2}$ be domains of dimensions $d_{1},d_{2}$
respectively, suppose that $pl>d_{2}$ or $p=1,l=d_{2}$. Then functions from
the Sobolev space $W_{p}^{l}(\Omega_{1}\times\Omega_{2})$ ($l$-th generalized
derivatives are summable with power $p$) are virtually continuous as functions
of two variables $x\in\Omega_{1}$, $y\in\Omega_{2}$. Embedding
$W_{p}^{l}(\Omega_{1}\times\Omega_{2})$ into $VC^{1}(\Omega_{1},K)$ is
continuous for any compact subset $K$ of the domain $\Omega_{2}$.

###### Proof.

Using the theorem of embedding of Sobolev space into continuous functions
(see, for instance, [2, 6]), we have the following estimate for functions
$h(y)\in W_{p}^{l}(\Omega_{2})$:

$\|h\|_{C(K)}\leq c(\Omega_{2},K)\|h\|_{W_{p}^{l}(\Omega_{2})}.$

Let $f(x,y)\in W_{p}^{l}(\Omega_{1}\times\Omega_{2})$ be a smooth function.
Set

$a(x):=\|f(x,\cdot)\|_{W_{p}^{l}(\Omega_{2})}.$

Then by Fubini’s theorem $a\in L^{1}(\Omega_{1})$ and

$\int|a|\leq
c(\Omega_{1},\Omega_{2})\|f(x,y)\|_{W_{p}^{l}(\Omega_{1}\times\Omega_{2})}.$

The following estimate holds on $\Omega_{1}\times K$:

$|f(x,y)|\leq\|f(x,\cdot)\|_{C(K)}\leq c(\Omega_{2},K)a(x).$

Summarizing this we have

$\|f\|_{VC^{1}(\Omega_{1},K)}\leq
c(\Omega_{1},\Omega_{2},K)\|f\|_{W_{p}^{l}(\Omega_{1}\times\Omega_{2})}.$ (2)

Each function in the class $W_{p}^{l}(\Omega_{1}\times\Omega_{2})$ is a limit
of a sequence of smooth functions, by (2) it is a limit in $VC^{1}$ as well. ∎

So, under conditions of this theorem we may integrate functions over
quasibistochastic measures. It generalizes usual theorems about traces on
submanifolds.

###  3.2 Nuclear operators in Hilbert space

It is well known that the space of nuclear operators in the Hilbert space
$L^{2}$ is a projective tensor product of Hilbert spaces. Their kernels are
measurable functions of two variables, which can hardly be described directly.
the following theorem claims that kernels of nuclear operators are virtually
continuous as functions of two variables. Note that kernels of Hilbert-Schmidt
operators are not in general virtually continuous.

######  Theorem 11.

Let $(X,\mu)$, $(Y,\nu)$ be standard spaces. the space of kernels of nuclear
operators from $L^{2}(X)$ to $L^{2}(Y)$ (with Schatten – von Neumann norm)
embeds continuously into $VC^{1}$.

It implies that such kernels may be integrated not only over diagonal when
$X=Y$, which is well known, but by bistochastic measures. But the space
$VC^{1}$ is wider than kernels of nuclear operators. If we look at $VC^{1}$ as
to the space of kernels of integral operators, it is not unitray invariant, on
the contrast to Schatten–von Neumann spaces. indeed, the definition of
$VC^{1}$ essentially uses known sigma-subalgebras, which do not have necessary
invariance. Close question is considered in [14].

###  3.3 Restrictions of metrics

The following problem was one of origins of this paper. Let $(X,\mu)$ be a
standard space with continuous measure. Assume that $\rho$ is an admissible
metric and $\xi$ is a measurable partition of $(X,\mu)$ with parts of null
measure (say, $\xi$ is a partition onto level sets of function which is not
constant on sets of positive measure). May we correctly restrict our metric (a
s a function of two variables) onto elements of this partition?

It is not immediately clear, since the metric is a priori just a measurable
function. But admissible metric is virtually continuous, and so for our goal
it suffices to define a bistochastic measure, onto which we have to restrict
it. Suppose for simplicity that $X=[0,1]^{2}$, $\mu$ is a Lebesgue measure,
$\xi$ is a partition onto vertical lines. Then we say about restriction of
virtually continuous function defined on $X^{2}=[0,1]^{4}$ onto three-
dimensional submanifold $\\{(x_{1},x_{2},x_{3},x_{4}):x_{1}=x_{3}\\}$. It is
easy to see that such a submanifold equipped by a three-dimensional Lebesgue
measure defines a bistochastic measure on $X\times X$.

##  4 Acknowledgements

We are grateful to L.Lovasz, who sent us his recent monograph [15], in which
close questions are discussed, and to A.Logunov for paying our attention to
the possibility of dual definition of the thickness.

## References

  * [1] L. V. Kantorovich. On the Translocation of Masses, Dokl. Akad. Nauk SSSR, 37, No. 78, 227229 (1942). English translation: Journal of Mathematical Sciences, 3-2006, Volume 133, Issue 4, pp 1381-1382.
  * [2] V. G. Mazýa. Sobolev spaces, Izdat. Leningrad. Univ., Leningrad (1985). English translation: Springer-Verlag, Berlin, (1985).
  * [3] M. Gromov. Metric Structure for Riemannian and Non-Riemannian Spaces. Birkhouser (1998).
  * [4] A. Vershik. The universal Uryson space, Gromov’s metric triples, and random metrics on the series of natural numbers. Uspekhi Mat. Nauk 53, No.5, 57-64 (1998). English translation: Russian Math. Surveys 53, No.5, 921-928 (1998).
  * [5] A. Vershik. Classification of measurable functions of several arguments, and invariantly distributed random matrices (in Russian). Funkts. Anal. Prilozh. 36, no.2, 12-28 (2002). English translation: Funct. Anal. Appl. 36, no.2, 93-105 (2002).
  * [6] R. A. Adams, J. J. .F Fournier. Sobolev spaces. Academic press, v. 140 (2003).
  * [7] A. Vershik. Three lectures on invariant measures and universality. In: Dynamics and Randomness II, A.Maass, S.Martinez, and J.San Martin (eds.), Kluwer Academic Publ., Netherlands, 2004, pp.199-228.
  * [8] A. Vershik. Random metric spaces and universality. Uspekhi Mat. Nauk 59, No. 2(356), 65-104 (2004). English translation: Russian Math. Surveys 59, No. 2, 259-295 (2004).
  * [9] A. Vershik. Polymorphisms, Markov processes, and quasi-similarity. Discrete Contin. Dyn. Syst. 13, No. 5, 1305-1324 (2005).
  * [10] A. Vershik. Vershik A. What does a generic Markov operator look like? Algebra i Analiz 17, No. 5, 91-104 (2005). English translation: St. Petersburg Mathematical Journal. 17, No. 5. p. 763-772. (2006)
  * [11] A. Vershik. Dynamic of metrics in measure spaces and their asymptotic invariants. Markov Processes and Related Fields, 16 No.1, 169-185 (2010).
  * [12] P. Zatitskiy, F. Petrov. Correction of metrics. Zapiski Nauchnykh Seminarov POMI, 390, 201-209 (2011). English translation: arXiv:1112.2380.
  * [13] K-T. Sturm. The space of spaces:curvature bounds and gradient flows on the space of metric measure space. (Preprint)
  * [14] M. Denker, M. Gordin. Limit theorem for von Mises statistics of a vrasure preserving transformations. arXiv:1109.0635v2
  * [15] L. Lovasz. Large networks and graph limits. Colloquium Publications, Vol. 60. (2012)
  * [16] A. Vershik. On classification of measurable functions of several variables, Zapiski Nauchn. Semin. POMI 403, 35-57 (2012). English version to appear in J. Math. Sci. 190, No. 3 (2013).
  * [17] F. Petrov, A. Vershik, P. Zatitskiy. Geometry and dynamics of admissible metrics in measure spaces. Central Europ.J.Math. 11, No.3, 379-400. (2013)
  * [18] A. Vershik. Long History of Monge-Kantorovich transportation problem. Mathem.Intellegencer, 35, No.4. (2013).


# The TELPERION survey for extended emission regions around AGN: a strongly-
interacting and merging galaxy sample

William C. Keel1,2 , Alexei Moiseev3,4, Roman Uklein3, and Aleksandrina
Smirnova3
1Dept. of Physics and Astronomy, University of Alabama, Box 870324,
Tuscaloosa, AL 35487, USA
2SARA Observatory, Embry-Riddle Aeronautical University, Daytona Beach FL
32114, USA
3Special Astrophysical Observatory, Nizhny Arkhyz, Russia
4Lomonosov Moscow State University, Sternberg Astronomical Institute,
Universitetsky pr. 13, Moscow 119234, Russia
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present the results of a search for Extended Emission-Line Regions (EELRs)
ionized by extant or recently-faded active galactic nuclei (AGN), using [O
III] narrowband imaging and spectroscopic followup. The sample includes 198
galaxies in 92 strongly interacting or merging galaxy systems in the range
$z=0.009-0.0285$. Among these, three have EELRs extended beyond 10 kpc in
projection from the nucleus detected in previous studies. We identify a single
new distant emission region, projected 35 kpc from UGC 5941. Our optical
spectrum does not detect He II, but its strong-line ratios put this in the
same class as securely-characterized EELR clouds. The nucleus of UGC 5941 is
dominated by recent star formation, preventing detection of any weak ongoing
AGN. Overall counts of distant EELRs in this and the previous TELPERION
samples give incidence 2–5% depending on galaxy and AGN selection, 20-50 times
higher than the Galaxy Zoo EELR survey with its higher surface-brightness
threshold and much larger input sample. AGN in interacting and merging systems
have an increased detection rate $12\pm 6$%, while none are detected around
noninteracting AGN. Some of these AGN are at luminosity low enough to require
additional X-ray or far-infrared information to tell whether the EELR
ionization level suggests long-term fading.

###### keywords:

galaxies: active – galaxies: interactions – galaxies: Seyfert

††pubyear: 2023††pagerange: The TELPERION survey for extended emission regions
around AGN: a strongly-interacting and merging galaxy sample–References

##  1 Introduction

Among active galactic nuclei (AGN), most types are characterised by strong
emission lines, whose ionization level distinguishes their spectra from those
of objects powered by star formation. In addition to the high-density broad-
line region and surrounding narrow-line region, some AGN also have extended
emission-line regions (EELRs) extending for kiloparsecs or tens of kiloparsecs
from the nucleus itself.

Objects described as EELRs in the literature are ionized directly or
indirectly by AGN, as distinct from other kinds of extended emission-line
regions such as starburst winds, widespread star formation, or shocks
occurring in merging galaxies. EELRs may be photoionized by radiation from the
AGN, or ionized by the effects of jets or broader outflows from radio-loud
AGN, generally including shocks and associated with emission-line Doppler
widths $>100$ km s-1 (Stockton, Fu, & Canalizo, 2006).

Photoionized EELRs offer the opportunity to learn about the AGN’s ionizing
radiation in directions or at times we cannot sample directly. They often
occur in pairs, triangular in projection and pointing to the AGN (ionization
cones, reviewed by Wilson 1996), which furnish strong evidence in favor of a
broad picture of circumnuclear obscuration producing much of the difference
between type 1 and 2 AGN (depending on whether the broad-line region is
obscured or not along our line of sight). A second facet of photoionized EELRs
was exemplified by the discovery of Hanny’s Voorwerp (Lintott et al., 2009)
during the initial Galaxy Zoo project (Lintott et al., 2008), where the
ionizing luminosity needed to match the EELR’s luminosity and ionization level
far exceed what is seen from the host galaxy’s nucleus directly. This provides
the possibility of sampling the AGN luminosity at times up to $10^{5}$ years
earlier than the direct view, giving the promise of retrieving the accretion
history of supermassive black holes over timescales not otherwise possible on
human timescales. The characterization of Hanny’s Voorwerp led to a followup
search by Galaxy Zoo volunteers, based on the distinctive colour of low-
redshift [O III] emission regions in $gri$ colour-composite renditions of
Sloan Digital Sky Survey (SDSS) images. This project confirmed another 19
distant EELRs, most found for the first time, and allowed a rough estimate
(0.2–2$\times 10^{5}$ years) of the duration of luminous accretion events for
these AGN (Keel et al., 2012). About 0.12% of the AGN examined were found to
show distant EELRs to surface-brightness levels apparent in SDSS data.

To complement the Galaxy Zoo searches, we have carried out a phased survey
called TELPERION111An acronym for Tracing Emission Lines Probing Extended
Regions Ionized by Once-active Nuclei, and recalling a past age of vanished
brilliance for readers of J.R.R. Tolkien’s stories of the First Age of Middle-
Earth.. This uses narrowband imaging of several hundred galaxies in a
convenient redshift range to seek emission clouds at lower surface brightness
than was detectable in the original SDSS images, out to projected distances in
principle as much as 250 kpc from the AGN hosts (so, for photoionized EELRs,
we could detect clouds tracing the luminosity of the AGN as much as $10^{5}$
years before the directly-observed epoch). TELPERION includes samples of
galaxies which are complete in various senses. The initial phase of the survey
(Knese et al., 2020) was comprised of Seyfert galaxies with known H I
properties, so it was clear which systems had enough neutral gas to produce
detectable [O III] emission. This phase uncovered one distant EELR, Mkn 1, out
of 26 AGN hosts. A second survey phase (Keel et al., 2022) targeted 111
luminosity-selected AGN hosts (and any companions), finding distant EELRs in
NGC 235 and NGC 5514. This phase also included merging galaxies from an
enhanced version of the “Toomre sequence" of ongoing mergers in our redshift
range, recovering the EELR previously found in NGC 7252 by Schweizer et al.
(2013).

Our primary interest in these surveys is identification of EELRs extending
beyond 10 kpc in projection from the AGN, which we term distant EELRs.
Compared to regions smaller and closer to the galaxy nuclei, these are less
likely to consist of the ordinary ISM of the host galaxy, allow us to probe
longer light-travel-time delays, and are easier to detect outside the bright
parts of host galaxies for better completeness of our samples. We also note
that the great majority of distant EELRs in both the Galaxy Zoo sample and
TELPERION are found around radio-quiet AGN (simply because most AGN are not
radio-loud objects with kpc-scale jets). Radio-loud objects have EELRs that
often reflect outflow associated with the jets or shocks as the jets encounter
the host galaxy ISM, rather than the more kinematically-quiescent AGN-
photoionized clouds we focus on.

A recurring pattern among the distant EELRs found in both the Galaxy Zoo and
TELPERION surveys has been that almost all are found in interacting or merging
galaxies, possibly because tidal features often include neutral gas that can
be observed when ionized by escaping UV continuum from the AGN (and such
features often depart from the host galaxy’s plane, reducing the absorption
along the path of the ionizing photons). We present here results of a further
phase of the TELPERION survey specifically aimed at interacting and merging
galaxies, and preliminary statistics of the occurrence of distant EELRs among
all four sample phases of the TELPERION survey.

Distance-dependent quantities are calculated using ${\rm H_{0}}=69.6$ km s-1
Mpc-1, $\Omega_{\rm M}=0.286$, and assuming flat geometry (Bennett et al.,
2014), with the Javascript calculator described by Wright (2006).

##  2 Sample Selection and Observations

###  2.1 Sample selection

To further examine the connection between distant EELRs and strong
interactions or mergers, we selected galaxies from the Darg et al. (2010)
sample, of galaxies having been flagged as mergers in the Galaxy Zoo project
(Lintott et al., 2008), which classified galaxies from data release 7 (DR7) of
the SDSS (Abazajian et al., 2009). This sample includes ongoing mergers,
tidally distorted interacting systems, and a small admixture of projected
pairs at substantially different distances (Keel et al., 2013). As in the
previous TELPERION phases, we used off-the-shelf narrowband filters to detect
[O III] $\lambda 5007$-Å emission (hereinafter simply [O III]) within the
range $z=0.0085-0.0285$, a range which includes 217 systems from the Darg et
al. (2010) catalog. We did not require the galaxy nucleus to be classified as
an AGN from optical spectra, but we did omit galaxies with central starbursts
as judged from SDSS spectra, to avoid emission from outlying star-forming
regions or starburst winds, not associated with an AGN. Following Stasińska &
Leitherer (1996), Fernandes, Leão, & Lacerda (2003), and Bergvall et al.
(2016), we omitted galaxy pairs where a non-AGN nuclear spectrum has H$\beta$
equivalent width EW(H$\beta)>20$ Å. This criterion helps make our EELR survey
more efficient, though we cannot apply it completely, since some pairs have
only a single SDSS spectrum available, and some galaxies’ spectra represent
off-nuclear knots rather than their nuclei. This left 95 systems, most with
two members. Of these, three systems are found to be projections of unrelated
and undistorted galaxies based on redshifts in later SDSS data releases, so we
omit these from our analysis: MCG +06-20-035, SDSS J114328.50+094947.2, and
SDS J140531.65-012147.9. IC 3935 includes a background system, but the
foreground disc is asymmetric across a wide radial range so we include it as
the aftermath of a strong interaction or minor merger. Our sample then
includes 92 strongly-interacting and merging systems containing 198 distinct
galaxies (although what counts as “distinct" in an ongoing merger is sometimes
a matter of individual judgment).

The Darg et al. (2010) sample of interacting and merging galaxies was selected
using classifications of galaxies in SDSS DR7. One system observed in our
sample, NGC 5541 was included as a photometric object (SDSS
J141631.80+393520.5) in SDSS DR7 but not DR9 (Ahn et al., 2012) and later,
apparently because of sky-subtraction issues due to the bright star HD 125111
projected 9′ away. A coordinate-based match cross-match of sample objects
between DR7 and DR9 would miss this one.

To help understand the content of the sample further, we classified the
nuclear spectra of the galaxies based on SDSS spectra and our previous
observations (Keel et al., 1985), for simplicity using the [O III]/H$\beta$-[N
II]/H$\alpha$ BPT diagram (Baldwin et al. 1981, Kewley et al. 2001, Kauffmann
et al. 2003) for emission-line spectra. The pipeline classifications for AGN
listed in the SDSS database use a linear approximation to the Seyfert+LINER
lower bound which falls above the maximum-starburst curves from Kewley et al.
(2001), and Kauffmann et al. (2003) everywhere, so we work from the SDSS line
fluxes when model-corrected for underlying stellar absorption (and apply
color-based corrections for H$\alpha$ and H$\beta$ absorption for our own
data). From emission lines, we find 68 star-forming objects, 32 LINERS, 11
Seyfert galaxies including objects near the Seyfert/LINER boundary, and 3
composite nuclei near the AGN lower boundary. Four nuclei show absorption
spectra with strong Balmer lines, indicating post-starburst systems. We find
37 quiescent systems, with the equivalent width of H$\alpha$ emission less
than 3 Å and no other emission detected except in some cases [O II]. No
optical spectra were available for 43 of the 198 galaxy nuclei in this sample.

###  2.2 Imaging observations

Our imaging procedures followed those in the previous phases of the TELPERION
survey (Knese et al. 2020, Keel et al. 2022). We searched for extended and
distant [O III] features using a filter with center wavelength 510 nm and
nominal half-transmission points at 505 and 515 nm (denoted F510), which
should shift blueward in a converging beam by no more than 4 Å in our
telescopes. Our sample includes galaxies where the stronger [O III] emission
line is observed within this range ($z=0.0085-0.0285$). Our narrowband images
were obtained with the remotely-operated 1-m telescopes of the SARA
observatory (Keel et al., 2017) at Kitt Peak (KP, 0.44″per pixel, field of
view 14.5′ square) and La Palma (the Jacobus Kapteyn Telescope, JKT; 0.34″per
pixel, field of view 11.6′ square). While explicit continuum subtraction often
has large residuals in star images due to PSF mismatch, we did obtain $V$
images for comparison, with one of those telescopes or the 0.6m SARA
instrument at Cerro Tololo (CT). Our survey F510 images typically reach a
surface brightness limit $1.0\times 10^{-16}$ erg cm-2 s-1 arcsec-2 in the
$\lambda 5007$ line. The $V$ continuum images were exposed to have at least
twice the signal-to-noise ratio of the narrowband data. As noted with the
previous TELPERION samples, use of [O III] for such a survey helps suppress
detections not only of star-forming regions relative to AGN-ionized gas, but
other emission clouds not ionized by AGN, including large-scale shocks in
merging systems and low-ionization material in the cores of galaxy clusters,
in which the strongest optical emission lines are typically H$\alpha$ and [N
II].

The imaging observations are documented in Table 1, with observing site and
UTC date for each observation. When poor image quality led us to repeat an
observation, only the higher-quality one is listed. Images were examined with
various levels of Gaussian and median smoothing in search of diffuse [O III]
structures. Potential diffuse [O III] clouds were reobserved for confirmation
before performing spectroscopic followup, to reduce time spent on false
positives near the detection threshold (effectively allowing the survey to
reach a deeper limiting surface brightness).

For the previously unknown distant [O III] cloud near UGC 5941, we obtained a
total of 13 one-hour exposures in F510 to examine its structure, at both KP
and the JKT.

Images were acquired for this project over the time frame from January 2017 to
May 2022. It was interrupted not only by shutdowns associated with
unavailability of onsite support personnel during the CoViD-19 pandemic
lockdowns at each site, but the Cumbre Vieja volcanic eruption on the island
of La Palma (September 2021 - January 2022) and the aftermath of the Contreras
wildfire at Kitt Peak (June 2022 - January 2023).

These images were not as confused by asteroid trails as the previous TELPERION
phase. However, attention to the order of targets during one night allowed us
to include the OSIRIS-REx asteroid probe in the $V$ image of the NGC 78 system
during an Earth swingby. We illustrate this, along with the $V$ continuum
image of this galaxy pair, in Fig. 1. Coordinates and spacecraft range were
calculated using the JPL Horizons ephemeris
service222https://ssd.jpl.nasa.gov/horizons/app.html#/ .

Figure 1: $V$-band continuum image of the galaxy pair NGC 78AB along with the
OSIRIS-Rex asteroid probe during Earth swingby, obtained with the Jacobus
Kapteyn Telescope. The main panel shows a combination of four 5-minute
exposures, combined with the unusual procedure of taking the maximum value at
each pixel location to retain the spacecraft trails. The four insets show the
trails during each individual exposure, and the right-hand inset shows the
galaxy pair at lower contrast to emphasize inner detail. The scale bar in each
case is 60″in length. The range to OSIRIS-Rex was 837,300 km in the middle of
this sequence.

###  2.3 Spectroscopic followup

The single candidate EELR newly found in this survey, near UGC 5941, was
observed spectroscopically using the SCORPIO-2 multimode instrument (Afanasiev
& Moiseev, 2011) on the 6-meter telescope (Bol’shoi Teleskop Azimutal’nyi,
BTA) at the Special Astrophysical Observatory of the Russian Academy of
Sciences. The SCORPIO-2 instrument was operated in its long-slit spectroscopic
mode, employing a 1″ slit with a usable length of 394″, oriented in position
angle 39∘ through the nucleus of UGC 5941. We obtained a 4500s exposure on
2023 March 15 UTC with the E2V 261-84 CCD detector (Afanasieva et al., 2023).
The spectrum covered the range 3930–8560 Å with 1.21 Å pixel sampling
(reciprocal dispersion). As well as the potential EELR, the spectrum records
fainter emission from a region just to the SW of the galaxy disc. Since the [S
II] lines overlap the spectral range affected by telluric B-band absorption, a
correction was applied based on the spectrum of a reference star. The
sensitivity of this spectrum detects a larger extent to the cloud than even
our deepest stacked direct image, showing [O III] for 9″along the slit and two
emission peaks.

We derive error and upper-limit estimates by adding fictitious lines at random
wavelengths while simultaneously fitting multiple Gaussian profiles in each
wavelength range (using the IRAF splot task), so these estimates include the
effects of line width as derived from the strong lines and nonlinear effects
of the fitting algorithm.

We also incorporate long-slit data from SCORPIO-2 in three slit positions
covering the emission filaments in NGC 5278/9, as described in Keel et al.
(2019).

###  2.4 Additional narrowband images

We make use of further narrowband [O III] images of two previously-detected
EELR systems, Mkn 266 and NGC 5278/9. On- and off-band images of NGC 5278/9
for [O III] were obtained with the MaNGaL tunable-filter system (Moiseev,
Perepelitsyn, & Oparin, 2020) at the 2.5 telescope of the Caucasus Mountain
Observatory333http://lnfm1.sai.msu.ru/kgo/main.php (Shatsky et al. 2020,
Potanin et al. 2017), with 2400s exposures for each. A scaled difference image
was used to isolate pure [O III] emission. For Mkn 266, we examine an [O III]
image produced by line fits through a data cube obtained with SCORPIO-2 in its
Fabry-Perot scanning mode (Moiseev, 2021), with total exposure 3600s.

Table 1: Sample galaxies and log of imaging observations SDSS name | $z$ | V date | V site | F510 date | F510 site | Other ID | Notes
---|---|---|---|---|---|---|---
SDSS J001755.39+002006.8 | 0.018 | 2017-11-16 | CT | 2022-01-20 | KP | MCG +00-01-059 |
SDSS J002027.50+005000.8 | 0.018 | 2017-09-20 | JKT | 2021-10-06 | KP | NGC 78 |
SDSS J014503.30-001805.4 | 0.018 | 2017-01-17 | JKT | 2020-12-21 | KP | UGC 1225 |
SDSS J020920.84-100759.1 | 0.013 | 2017-09-20 | CT | 2017-12-20 | KP | NGC 833 |
SDSS J024213.22-082934.9 | 0.024 | 2017-09-28 | JKT | 2020-01-01 | KP | KUG 0239-087 |
SDSS J032045.32-010314.2 | 0.021 | 2017-10-31 | CT | 2018-01-17 | KP | UGC 2691 |
SDSS J075121.01+501409.8 | 0.024 | 2017-11-13 | KP | 2021-12-08 | KP | UGC 4052 |
SDSS J080724.81+065147.5 | 0.015 | 2017-11-20 | CT | 2021-02-10 | KP |  |
SDSS J080945.62+255253.2 | 0.025 | 2018-03-13 | JKT | 2018-03-12 | KP | IC 2229 |
SDSS J081312.31+402510.3 | 0.026 | 2018-03-15 | JKT | 2018-12-13 | KP |  |
SDSS J082724.89+233402.1 | 0.019 | 2018-03-15 | JKT | 2019-01-14 | KP |  |
SDSS J083031.67+181221.9 | 0.027 | 2017-11-20 | CT | 2022-03-31 | KP |  |
SDSS J083824.01+254516.2 | 0.018 | 2019-03-06 | JKT | 2019-03-04 | KP | NGC 2623 |
SDSS J084740.18+255336.6 | 0.022 | 2018-01-17 | KP | 2022-04-05 | KP | UGC 4597 |
SDSS J084924.19+190427.4 | 0.013 | 2018-05-15 | JKT | 2020-12-21 | KP | NGC 2673 |
SDSS J085601.86+421624.6 | 0.028 | 2018-11-15 | JKT | 2020-02-12 | KP | MCG +07-19-002 |
SDSS J090159.93+352031.9 | 0.026 | 2019-12-17 | JKT | 2020-02-12 | KP |  |
SDSS J090554.49+471045.5 | 0.027 | 2021-03-17 | JKT | 2021-03-17 | JKT | UGC 4765 |
SDSS J091029.79+203344.3 | 0.028 | 2019-12-27 | JKT | 2020-12-14 | JKT |  |
SDSS J091646.06+454849.6 | 0.026 | 2019-02-04 | JKT | 2020-12-21 | KP | MCG +08-17-066 |
SDSS J092229.43+502548.4 | 0.027 | 2019-11-25 | JKT | 2021-02-10 | KP | MCG +08-17-087 |
SDSS J092540.12+221920.8 | 0.025 | 2019-11-26 | JKT | 2021-02-16 | KP | CGCG 121-087 |
SDSS J093456.42+384441.9 | 0.015 | 2019-11-26 | JKT | 2021-03-10 | KP |  |
SDSS J093634.03+232627.0 | 0.028 | 2021-03-17 | JKT | 2021-03-17 | JKT | CGCG 122-029 |
SDSS J093939.93-001407.0 | 0.016 | 2021-03-12 | JKT | 2020-02-24 | KP | NGC 2951 |
SDSS J094751.71+084950.8 | 0.018 | 2019-11-26 | JKT | 2021-03-10 | KP | MCG +02-25-036 |
SDSS J095648.41+530853.8 | 0.025 | 2018-03-13 | JKT | 2019-04-24 | KP | VV 740 |
SDSS J100358.83+221633.8 | 0.020 | 2018-05-15 | JKT | 2020-12-21 | KP | UGC 5420 |
SDSS J100549.83+003800.0 | 0.021 | 2019-12-27 | JKT | 2020-01-06 | KP | IC 590 |
SDSS J102424.36+280143.3 | 0.021 | 2019-12-27 | JKT | 2020-02-12 | KP | NGC 3232 |
SDSS J102743.61+011138.2 | 0.022 | 2019-12-27 | JKT | 2021-02-16 | KP |  |
SDSS J103635.16+374228.6 | 0.023 | 2019-12-27 | JKT | 2021-03-17 | JKT |  |
SDSS J104503.59+002603.6 | 0.026 | 2019-12-27 | JKT | 2020-12-14 | JKT |  |
SDSS J104646.71+030942.8 | 0.022 | 2019-12-27 | JKT | 2022-04-04 | KP |  |
SDSS J105021.59+412750.5 | 0.024 | 2018-03-13 | JKT | 2019-05-04 | KP | UGC 5941 | New [O III] cloud
SDSS J105143.63+510119.7 | 0.025 | 2019-12-27 | JKT | 2020-02-24 | KP | NGC 3406 |
SDSS J105237.39+080024.1 | 0.022 | 2021-03-12 | JKT | 2022-01-10 | KP | ALFALFA 5-358 |
SDSS J105738.52+410317.6 | 0.025 | 2020-01-14 | JKT | 2021-12-08 | KP |  |
SDSS J111809.78+302424.3 | 0.026 | 2021-03-12 | JKT | 2021-02-10 | KP | UGC 6314 |
SDSS J113218.80+555540.8 | 0.021 | 2020-01-14 | JKT | 2021-12-09 | KP |  |
SDSS J113240.25+525701.4 | 0.026 | 2018-05-15 | JKT | 2018-12-13 | KP | Mkn 176 |
SDSS J114044.47+222646.9 | 0.024 | 2018-01-17 | KP | 2022-03-31 | KP | NGC 3808 |
SDSS J114222.33+084611.5 | 0.022 | 2018-05-15 | JKT | 2022-04-05 | KP | IC 720 |
SDSS J115805.22+275244.0 | 0.011 | 2021-03-12 | JKT | 2021-02-10 | KP | NGC 4004 |
SDSS J120445.19+311132.9 | 0.025 | 2021-03-12 | JKT | 2021-03-10 | KP |  |
SDSS J120607.51+325340.9 | 0.027 | 2020-01-14 | JKT | 2020-02-24 | KP | IC 2993 |
SDSS J121155.07+403918.2 | 0.022 | 2021-03-12 | JKT | 2022-02-03 | KP |  |
SDSS J121535.84+281039.4 | 0.022 | 2018-03-13 | JKT | 2022-04-05 | KP | NGC 4211 |
SDSS J121917.19+120058.2 | 0.026 | 2021-04-05 | CT | 2022-02-28 | KP | IC 3147 |
SDSS J122257.72+103253.9 | 0.026 | 2020-03-16 | CT | 2021-02-10 | KP | NGC 4320 |
SDSS J122814.15+442711.7 | 0.023 | 2021-03-12 | JKT | 2021-03-10 | KP | UGC 7593 |
SDSS J123108.37+003649.3 | 0.023 | 2021-04-05 | CT | 2022-03-31 | KP | NGC 4493 |
SDSS J123634.26+111419.9 | 0.008 | 2022-05-22 | KP | 2022-03-27 | KP | NGC 4567/8 |
SDSS J124109.46+293219.1 | 0.024 | 2021-03-12 | JKT | 2022-01-31 | KP |  |
SDSS J124549.51+404904.2 | 0.024 | 2021-12-08 | KP | 2021-12-08 | KP |  |
SDSS J124610.14+304354.9 | 0.022 | 2021-03-17 | JKT | 2021-02-16 | KP | NGC 4676 |
SDSS J125649.89+481800.4 | 0.028 | 2019-06-28 | KP | 2022-04-04 | KP | NGC 4837 | UGC 8068
SDSS J125731.95+282836.9 | 0.023 | 2019-06-19 | KP | 2022-04-04 | KP | NGC 4841 |
SDSS J125818.23+290743.6 | 0.026 | 2018-07-05 | JKT | 2022-04-04 | KP | IC 3935 |
SDSS J125821.70+280855.5 | 0.026 | 2022-04-05 | KP | 2022-02-11 | JKT | NGC 4851 |
SDSS J125940.10+275117.7 | 0.013 | 2022-05-01 | KP | 2022-02-28 | KP |  |
SDSS J130004.24+280918.3 | 0.022 | 2022-05-02 | JKT | 2022-02-03 | KP |  |
SDSS J130125.26+291849.5 | 0.023 | 2019-06-28 | KP | 2018-05-15 | KP | NGC 4922 |
Table 2: continued

Sample galaxies and log of imaging observations  Table 3:  SDSS name $z$ V
date V site F510 date F510 site Other ID Notes SDSS J130147.06+290435.9 0.023
2022-05-02 JKT 2022-03-27 KP SDSS J130702.74+133814.4 0.027 2021-03-12 CT
2022-01-31 KP KPG 365A SDSS J132035.40+340821.7 0.023 2018-06-14 KP 2021-02-21
KP IC 883 SDSS J132652.11-023436.6 0.012 2021-04-15 CT 2022-03-31 KP SDSS
J133525.40+332641.6 0.024 2020-07-24 JKT 2022-04-04 KP SDSS
J133817.77+481641.0 0.028 2021-03-17 JKT 2021-03-17 JKT Mkn 266 Keel et al.
(2012) cloud SDSS J134139.62+554014.3 0.025 2015-03-27 . KP 2015-03-27 KP. NGC
5278 Keel et al. (2019) cloud SDSS J135837.97+372528.2 0.012 2021-03-17 JKT
2020-02-21 KP NGC 5394/5 SDSS J140037.96-025422.7 0.025 2021-03-10 KP
2021-03-10 KP IC 968 SDSS J140141.38+334936.5 0.026 2020-07-24 JKT 2021-05-07
KP NGC 5421 SDSS J140821.09+354221.3 0.028 2022-04-05 KP 2022-02-03 KP SDSS
J140834.22+091147.7 0.023 2021-04-15 CT 2022-03-31 KP SDSS J141338.69+073935.8
0.024 2021-03-17 JKT 2021-03-17 JKT NGC 5514 TELPERION cloud SDSS
J141342.00+081316.3 0.025 2021-04-15 CT 2022-02-03 KP UGC 9103 SDSS
J141422.99+025831.5 0.026 2020-03-16 CT 2022-02-28 KP SDSS J141631.80+393520.5
0.026 2020-07-23 JKT 2022-02-03 KP NGC 5541 DR7, not in DR12 SDSS
J141702.52+363417.7 0.010 2020-07-24 JKT 2022-03-27 KP NGC 5544/5 SDSS
J141814.94+342659.8 0.028 2020-07-23 JKT 2022-03-03 KP SDSS
J142409.12+121531.1 0.027 2021-04-15 CT 2022-20-28 KP SDSS J144042.83+032755.5
0.027 2019-06-03 CT 2022-03-27 KP NGC 5718 SDSS J144514.14+384344.1 0.015
2018-03-13 JKT 2022-04-04 KP NGC 5752 SDSS J150448.33+410016.1 0.028
2020-07-23 JKT 2022-02-11 JKT SDSS J151100.89+233633.3 0.017 2020-07-24 JKT
2022-03-27 KP MCG +04-36-024 SDSS J152607.95+414033.8 0.009 2020-07-21 JKT
2019-07-03 KP NGC 5929/30 SDSS J153926.06+245636.9 0.023 2020-07-23 JKT
2019-05-04 KP Mkn 860A SDSS J155912.54+204548.0 0.015 2020-07-21 JKT
2022-04-05 KP NGC 6027 SDSS J162946.50+405229.5 0.028 2017-09-28 JKT
2019-07-03 KP SDSS J215507.85-004613.0 0.027 2021-10-06 KP 2021-10-06 KP SDSS
J235410.08+002258.2 0.026 2021-10-06 KP 2021-10-06 KP NGC 7783

##  3 Results: EELRS in a large interacting and merging sample

This survey recovered previously-known distant EELRs in three systems, and one
probable new one. The three are Mkn 266 (with the highest-surface-brightness
EELR from the Galaxy Zoo sample, Keel et al. 2012), the emission filaments
beyond the stellar discs of NGC 5278/9 (Keel et al., 2019), and the very
distant cloud in the tidal tails of NGC 5514, found in the TELPERION AGN-host
sample (Keel et al., 2022). The new [O III] structure was found near UGC 5941
(see Fig. 2), at projected separation 70″or 35 kpc from the nucleus. The BTA
spectrum in Fig. 3 shows ratios of the strong emission lines that place it in
the AGN section of common classification diagrams, although its signal-to-
noise ratio is not sufficient to detect He II $\lambda 4686$. The upper limit
to the ratio He II $\lambda 4686$/H$\beta<0.25$ is high enough to be
consistent with the values seen on other EELRs (Keel et al. 2012, Knese et al.
2020, Keel et al. 2022). This line ratio has proven important in
distinguishing the ionization sources of gas at low metallicity, where the AGN
and star-formation loci can overlap in the most common BPT diagram (Baldwin et
al. 1981, Kewley et al. 2001, Kauffmann et al. 2003), as discussed by Groves,
Heckman, & Kauffmann (2006). It lies close in the BPT diagrams to previously-
identified EELRs with He II detections.

Properties measured from the spectrum of the UGC 5941 cloud are listed in
Table 2, as are measures of weaker and more diffuse emission just outside the
galaxy disc along the slit. The redshifts are slightly smaller than that we
measure for the galaxy nucleus, although the radial-velocity differences are
barely significant (listed under $v_{\rm offset}$ in the table, in the sense
object minus nucleus).

Table 2: Properties of UGC 5941 clouds Property | Cloud | Inner Emission
---|---|---
$\alpha_{2000}$ | 10:50:17.69 | 10:50:20.00
$\delta_{2000}$ | +41:26:56 | +41:27:36
$z$ | $0.02384\pm 0.00005$ | $0.02363\pm 0.00035$
$v_{\rm offset}$, km s-1 | $-34\pm 17$ | $-97\pm 105$
[O III] $\lambda 5007/$H$\beta$ | $4.6\pm 0.5$ | $>3,6$
[N II] $\lambda 6583/$H$\alpha$ | $0.46\pm 0.09$ | $0.19\pm 0.12$
[O I] $\lambda 6300/$H$\alpha$ | $0.18\pm 0.05$ | $<0.18$
H$\alpha$/H$\beta$ | $4.0\pm 1.0$ | $>5$
[S II] $\lambda\lambda 6717,6731$/H$\alpha$ | $<0.12$ | $<0.11$
Figure 2: Narrow- and broad-band images showing the [O III] cloud detected
near UGC 5941 (the galaxy pair at upper left). The area shown in each panel
spans 208″N-S by 242″E-W, with north at the top as shown. Each image has been
smoothed by a Gaussian with FWHM=0.65″for display. The upper panels compare
$V$ (predominantly continuum) and F510 images with comparable linear stretch,
showing the [O III] cloud to the SW and fainter structure just outside the
bright disc. At lower left, the $V$ image is shown with a log scale to reveal
the spiral pattern of UGC 5941, while the F510 image is repeated at lower
right with annotation including the location of the spectrograph slit.

###  3.1 Emission-line ratios and classifications

Common tools for understanding the energy input to gaseous nebulae, especially
for galactic nuclei, are the BPT diagrams (Baldwin et al., 1981) comparing
ratios of strong emission lines, as revised by later empirical and modeling
studies (Veilleux & Osterbrock 1987, Kewley et al. 2001, Kauffmann et al.
2003). The classic BPT classification diagrams are shown in Fig. 4 for the
various phases of the TELPERION survey, highlighting the objects appearing in
this work, and for the Galaxy Zoo EELR clouds, including Hanny’s Voorwerp near
IC 2497 and sharing Mkn 266 in common with the sample observed for this paper.
Only for UGC 5941 is there not a detection of He II at intensity
$>0.1$H$\beta$ to further support the identification of AGN ionization; its
cloud lies in the AGN region, above the maximum-starburst lines shown. The
Galaxy Zoo objects as a group have higher excitation (larger values of [O
III]/H$\beta$) than the TELPERION clouds, attributable to higher ionization
parameter and reflecting the deeper surface-brightness threshold of the
TELPERION narrowband survey compared to broad-band selection from SDSS images.

Figure 3: Diagnostic sections of the BTA long-slit spectrum of UGC 5941 and
its associated emission cloud. The nucleus, summed over a 1 ″$\times
2$″region, is scaled down as indicated. The distant cloud spectrum is summed
over a region 1″$\times 7.2$″, and the inner emission region is summed over
$1\times 7.6$″. Wavelengths of two weak or undetected lines which are
important in classification are marked - He II $\lambda 4686$ in the blue
segment and [O I] $\lambda 6300$ in the red segment. The [S II] doublet
observed near 6870 Å falls in a wavelength region so contaminated by night-sky
residuals that we can derive only upper limits for the cloud and inner
emission region. Figure 4: BPT emission-line classification diagrams for the
EELRs in the Galaxy Zoo sample and the varioos phases of the TELPERION survey.
Triangles indicate the Galaxy Zoo objects with red spectral coverage from Keel
et al. 2012, squares mark objects from the earlier TELPERION phases and
circles indicate objects from the sample observed for this work. The maximum-
starburst line is adopted from Kauffmann et al. 2003 (Ka03), and also shown
according to Kewley et al. 2001 (Ke01) for the leftmost diagram, with their
recommended uncertainty of $\pm 0.1$ dex. Separate points are shown for three
of the large filaments in the NGC 5278/9 system. We have taken values for NGC
7252 from Schweizer et al. 2013 and for Mkn 266 from Keel et al. 2012. The
right end of the arrow in the [S II] diagram marks the upper limit for UGC
5941. Error bars for the Galaxy Zoo sample are representative, from objects
with multiple observations.

###  3.2 Incidence of distant EELRs

A straightforward way to estimate the incidence of distant EELRs is to
consider the fractions of galaxies and known AGN hosts with such objects in
the various phases of the TELPERION survey. These include a sample of Seyfert
galaxies with H I detections (H I AGN in Table 3), so that it would be known
where any ionized structures lie with respect to extended neutral gas (Knese
et al., 2020); an extended version of the Toomre merging sequence (Toomre
seq), a sample of AGN hosts selected on luminosity (Keel et al., 2022); and
the present work selecting a larger sample of strongly interacting and merging
galaxies (Mergers). For completeness, we list the number of galaxies in each
sample (Targeted) and the number of galaxies examined for EELRs including any
additional comparably luminous galaxies at matching redshifts (All); no EELRs
appeared near these additional galaxies.

Simple tallies of the distant EELRs we have detected in the various TELPERION
samples are given in Table 3, where values for combined samples account for
galaxies in common between samples to avoid double-counting. These occur
because the sample selections for various phases of the TELPERION survey used
different kinds of criterion, and some galaxies satisfy more than one so were
included in multiple samples. This interacting and merging sample selected in
this work overlaps with the Toomre-sequence galaxies from Keel et al. (2022)
in the systems NGC 2623, NGC 4676, and IC 883, and has NGC 5394/5, NGC 5514,
and UGC 7064 in common with the AGN-host sample. In assessing the detection
statistics, we note further overlaps between the TELPERION samples as follows:
The H I AGN list (Knese et al., 2020) and AGN-host list from Keel et al.
(2022) share the four systems UGC 3995, NGC 7469, NGC 7591, and NGC 7682.

We consider the numbers of galaxies, systems, and Seyfert galaxies. Notably,
for the Toomre sample, the EELR-to-Seyfert galaxy ratio is undefined due to
the unique case of NGC 7252, which has an EELR without directly-observed
nuclear activity; (Schweizer et al., 2013). Various catalog sources differ in
their thresholds for listing Seyfert nuclei as distinct from some LINERs,
especially among the heterogenous sources used in compiling the VCV AGN
catalog (Véron-Cetty & Véron, 2010) used in constructing the AGN-host sample.
Our values include the UGC 5941 cloud as AGN-ionized, based on its location in
Fig. 4. Uncertainties in the occurrence rates for distant EELRs (EELRs/galaxy
and EELRs/Seyfert) use the binomial expression444where the 1-$\sigma$
uncertainty is $(\frac{p(1-p)}{N})^{1/2}$ for sample size $N$ and fractional
occurrence $p$., since the number of galaxies observed was set independently
of the number of cloud detections.

LINERs counted here all have strong enough H$\alpha$ emission (equivalent
width $>3$ Å ) to be in the AGN family, except for shocked gas in ongoing
mergers as in NGC 5514 (Lípari et al., 2004) and NGC 6240, although even in
this very dusty ultraluminous infrared galaxy (ULIRG) system there is evidence
for AGN photoionization in some directions (Medling et al., 2021).

Table 3: Galaxies and Distant EELRs in TELPERION Samples Sample | Galaxies: | Nuclei: | EELRs | EELR/galaxy | EELR/Seyfert | Reference
---|---|---|---|---|---|---
| Targeted | All | Seyfert | LINER |  |  |  |
H I AGN | 26 | 34 | 21 | 5 | 1 | $0.029\pm 0.028$ | $0.048\pm 0.041$ | Knese et al. (2020)
Toomre seq | 17 | 21 | 0 | 9 | 1 | $0.047\pm 0.046$ | — | Keel et al. (2022)
AGN hosts | 120 | 185 | 87 | 30 | 2 | $0.011\pm 0.007$ | $0.023\pm 0.013$ | Keel et al. (2022)
Mergers | 92 | 198 | 9 | 31 | 4 | $0.020\pm 0.010$ | $0.44\pm 0.05$ | This work
Toomre seq+merger | 106 | 215 | 9 | 35 | 5 | $0.023\pm 0.010$ | $0.56\pm 0.17$ |
H I AGN+AGN host | 141 | 211 | 103 | 32 | 3 | $0.014\pm 0.008$ | $0.029\pm 0016$ |
Interacting AGN | 34 | 34 | 34 | 40 | 5 | — | $0.12\pm 0.06$ |
All | 241 | 415 | 111 | 71 | 7 | $0.017\pm 0.006$ | $0.064\pm 0.023$ |

Whether counting per galaxy or per system, and whether we consider only
interacting and merging galaxies or wider samples, we find distant EELR clouds
near 2-4% of them. This contrasts with the Galaxy Zoo sample taken from a much
larger set of galaxies (roughly 16,000 potential AGN hosts) but with a
surface-brightness limit in [O III] emission several times higher. That survey
found 19 clouds for a fraction of order 0.001. The redshift bounds for the
samples in the TELPERION survey enforce effective limits on galaxy luminosity;
for the previous phase of the survey, AGN hosts were observed only if brighter
than absolute magnitude $-20$. We find occurrence fractions in TELPERION 20-40
times higher than the Galaxy Zoo sample, which may suggest that the number of
EELRs climbs steeply toward lower surface brightness, so that even deeper
imaging surveys might prove very fruitful.

Our results continue the pattern of distant EELRs occurring preferentially in
interacting and merging systems, as was seen in the Galaxy Zoo objects. Among
galaxies with Seyfert nuclei in the Toomre sequence sample and the merger
sample from this paper, roughly half ($57\pm 17$% ) show distant EELRs. We can
extend the statistics slightly by including interacting AGN (Seyferts and
LINERs) from the AGN-host and H I AGN samples, taking the labels for tidal
disturbance from Table 7 of Keel et al. (2022) and those listed by Kuo et al.
(2008) in their Group I, where H I features trace tidal interactions.
Excluding duplications with the other TELPERION samples, this addition
comprises 25 Seyfert galaxies, 5 LINERs, and two EELRs near Seyfert galaxies.
As shown in the “Interacting AGN" entry in Table 3, this category includes all
but one of the EELRs found in the whole TELPERION program. EELRs have been
detected near $12\pm 6$% of this combined sample (and none of the
noninteracting AGN observed).

###  3.3 Galaxies and AGN with distant EELRs

Combining the results of the four TELPERION phases, it is appropriate to
compare the nuclei and clouds of each of the 7 systems with distant EELRs.
Fig. 5 provides a visual comparison of the TELPERION EELRs, including UGC
5941, at the same scales in linear size and [O III] surface brightness. These
systems are discussed starting with the four found in this paper’s merger
sample.

Mkn 266 (NGC 5256) is part of a merging system, whose [O III] clouds have the
highest surface brightness among detections in the Galaxy Zoo survey (Keel et
al., 2012). With far-IR luminosity $10^{45}$ erg s-1, it qualifies as a
luminous infrared galaxy (LIRG). Both nuclei show Sy 2 spectra in the optical
range; this is one of the rare local examples of a dual AGN, with components
separated by 5.6 kpc in projection (Mazzarella et al., 2012). This, together
with the ongoing merger and consequent deep blending of the nuclei in far-IR
observations, makes it difficult to apportion the luminosity between the two
galaxies to tell whether one nucleus dominates the photoionization. Further,
an ongoing starburst accounts for roughly half of the far-IR (and therefore
bolometric) luminosity (Wang et al. 1997, Mazzarella et al. 2012). Mkn 266 is
part of the merger sample reported in this paper, and also appeared in the
Galaxy Zoo sample (Keel et al., 2012) due to the very high surface brightness
of its EELR filaments.

NGC 5278/9 (Arp 239, VV 19, Mkn 271) form a strongly-interacting pair of
spirals, which Vorontsov-Velyaminov (1977) called the “Telephone Handset"
system, noting a now-dated resemblance. Kinematic studies show the two
distinct rotating discs which are prominent morphologically, but departures
from circular coplanar motion are locally almost as large as the circular
velocities (Klimanov, Reshetnikov, & Burenkov 2002, Repetto et al. 2010.)
Simulations suggest that this combination, and the lack of bright tidal tails
seen in the starlight, mark a galaxy pair on an initial close approach before
the main bodies spiral inward toward merger. Both nuclei have similar optical
spectra, near the Seyfert/LINER boundary in BPT diagrams depending on aperture
size. Each is accompanied by highly-ionized filaments extending to projected
distances 28-58″from the nuclei (Keel et al., 2019); the spectroscopic values
we use here were obtained from new measurements on the data obtained for that
paper. Here again, the far-IR data blend both galaxies, and assessing how much
AGN radiation may be reradiated by dust is further complicated by the evident
levels of star formation in the discs. We have explored the IR characteristics
of this system using the Spitzer data obtained in the Star Formation Reference
Survey (SFRS; Ashby et al. 2011). The SFRS project used the consistent
continuum slope of a wide range of stellar populations across the mid-infrared
bands, from 3.6-8$\mu$m and therefore well longward of the generic 1.6$\mu$m
peak in their spectral energy distributions, to generate images which largely
remove the direct light of stars at 8$\mu$m, by subtracting an appropriately
scaled version of the 3.6$\mu$m image. The resulting image, their “NS"
combination, is especially powerful in isolated both warm dust in star-forming
regions and emission from AGN. This technique applied to NGC 5279 shows a
central source at 8$\mu$m that is 4 times as bright as the core of NGC 5278,
in contrast to their more equal fluxes in narrow (optical) emission lines
(where NGC 5279 is brighter in [O III] by a factor 1.5).

NGC 5514 is an advanced merger with two prominent tidal tails. The EELR,
initially reported by Keel et al. (2022), is near the tip of one tail, at the
extraordinary projected distance 75 kpc. Detailed analysis of that work’s
optical spectra in the main body, including the presence of multiple
superimposed velocity components, suggests that shock ionization is important
here, so that scaling from the emission-line luminosity will overestimate the
AGN luminosity. Integral-field spectra suggest that off-nuclear starburst
regions and associated outflows dominate the shock emission (Lípari et al.,
2004), rather than the large-scale shocks sometimes found in ongoing mergers.
Unlike most of the other LINER systems we deal with, in this case the line
ratios must largely reflect shocks. In contrast, the very distant EELR is in a
region near the end of a stellar tail where large-scale cloud collisions are
unlikely, shows small line widths and notably higher ionization levels than
gas near the galaxy center, all of which suggest photoionization by a
(possibly faded) AGN. The direct emission signature of such an AGN is lost in
the complex environment of other processes near the galaxy nuclei, providing
us only with an upper limit to AGN luminosity from available data.

UGC 5941 is an M51-type interacting pair, with a smaller companion projected
25″away near the tip of a tidal arm; the pair is accompanied by a similarly
luminous edge-on spiral (SDSS J105008.59+412638.8) projected 162″from UGC 5941
at a similar redshift.

Mkn 1 (NGC 449) shows an opposing pair of ionized clouds projected $\approx
13$ kpc from the nucleus found in the initial H I AGN phase of TELPERION
(Knese et al., 2020). One of these clouds has spectroscopic data of quality
sufficient to show a high He II $\lambda 4686$/H$\beta$ ratio indicating AGN
photoionization. The host galaxy is in a pair with the spiral NGC 451 at
projected separation 109″(equivalent to 35 kpc). Kuo et al. (2008) find a
common H I envelope including a tidal bridge and tails, suggesting that the
asymmetry and kinematic disturbance of the H I in the disc of Mkn 1 (also
noted by Omar et al. 2002 from GMRT data) have been caused by this
interaction. The nucleus shows a typical Sy 2 optical spectrum.

NGC 235, also designated NGC 235A in NED, is part of a tidally-distorted
interacting triple including NGC 232 and the smaller galaxy NGC 235B, seen
projected against the disc of NGC 235. The EELR (Keel et al., 2022) lies
between NGC 235 and NGC 232, whose nuclear optical spectrum shows star
formation without a significant AGN contribution.

NGC 7252, part of the original Toomre sequence, is a merger remnant in an
advanced state with a single nucleus, small-scale spiral pattern with star-
forming regions, and extensive tidal tails. Its EELR was reported by Schweizer
et al. (2013), and recovered in the TELPERION merging-sequence sample (Keel et
al., 2022). The ionized gas roughly follows one of the tidal tails over a
radial range 5-13 kpc from the nucleus. This system is notable for having no
detected AGN, furnishing a very strong case for essentially complete shutdown
of accretion luminosity over $1-2\times 10^{4}$ years.

Since many of the distant EELRs from the Galaxy Zoo sample are so bright as to
suggest that the AGN has faded over $10^{4}-10^{5}$ years, we consider the
available data for the TELPERION objects to see whether some show evidence for
fading. (AGN brightening is more difficult to demonstrate, since it has the
same energy-balance signature as lack of extended neutral gas or greater-than-
average AGN obscuration). For the luminous Galaxy Zoo AGN examined by Keel et
al. (2012), a simple picture could be used to bound the ionizing luminosity,
in which all ionizing radiation leaving the nucleus either ionizes observable
gas (estimated using scaling relations from [O III] luminosity) or is absorbed
and reradiated by dust, so the mid- and far-infrared luminosities are proxies
for the remaining fraction of ionizing radiation emitted from the AGN. As we
find EELRs associated with weaker (and sometimes unobservably faint) AGN,
these assumptions become progressively less realistic. The contribution to
grain heating from star formation becomes relatively more important for lower
AGN luminosity, and a larger fraction of the observed infrared fluxes for
lower angular resolution. Likewise, lower-luminosity AGN may have signatures
in the optical spectra that cannot be reliably separated from central star
formation using only the strong emission lines incorporated in the BPT
diagrams. The TELPERION objects are clearly in this more complex regime.

This complexity may be seen from the data in Table 4, which includes
information on the implied [O III] luminosity of each nucleus, and its
bolometric luminosity under the assumption that the entire far-infrared output
comes from the nucleus, as was done for the Galaxy Zoo sample. Here,
L${}_{\rm[O~{}III]}$ is the luminosity of the nucleus in the [O III] $\lambda
5007$ line, while the bolometric luminosity LBol combines contributions traced
by [OIII] and the far-infrared luminosity. All these values of Lion,cloud are
in fact lower limits, calculated for complete absorption of the incident
ionizing continuum, so the more realistic case of incomplete absorption
increases the required luminosity of the AGN. The column labelled “Ratio" is
the ratio of this implied bolometric output to the ionizing luminosity
Lion,cloud needed to ionize the cloud at the observed H$\alpha$ surface
brightness and (projected) distance from the nucleus, again estimated
following Keel et al. (2012); values smaller than unity would indicate that
the AGN has faded over the relevant light-travel time. To the extent that
surrounding star-forming regions produce additional far-IR emission within the
large beams of available FIR data, this ratio must be an overestimate,
sometimes by a very large factor. From these data, the AGN in NGC 5278/9, NGC
5514, and NGC 7252 show evidence of fading over timespans 0.9, 2.4, and
0.4$\times 10^{5}$ years, respectively. Mkn 1 is consistent with a constant
luminosity obscured along our line of sight, while the situations in Mkn 266
and NGC 235 remain ambiguous.

The case of the UGC 5941 EELR cloud illustrates some of the limits on this
energy-balance approach for galaxies so distant that existing mid- and far-
infrared data do not separate emission from the nucleus and disk of the host
galaxy. The AGN is faint enough to be lost in the star-forming signatures of
the galaxy both spectroscopically and in integrated infrared flux. Even if all
the IR output of the galaxy is assigned to reprocessed AGN radiation, the
cloud ionization still somewhat exceeds the level powered by an unobscured
line of sight from the cloud to the AGN, accounting for a fraction $<0.6$ of
it from Table 4). UGC 5941 is therefore a candidate for modest fading over
$\approx 10^{5}$ years. The optical spectrum from the nucleus shows star
formation, securely below the starburst dividing curves in all three BPT
diagrams, with [N II]/H$\alpha$=0.48, [O III]/H$\beta$=0.21. These ratios
suggest that only a small fraction $<0.05$ of the H$\alpha$ or H$\beta$
emission at the nucleus comes from an unobscured AGN, based on a simple mixing
model of two components typical of high-metallicity star-forming regions in
galaxy nuclei and of Seyfert-galaxy line ratios.

We can start to address the limitations of IR sky survey data, improving
luminosity estimates for low-luminosity AGN in star-forming host galaxies, for
NGC 5278/9, using the Spitzer data from 3.6-24 $\mu$m obtained by the SFRS
project (Ashby et al., 2011). This exercise sets tighter limits on the IR
output from the nuclei - at 8$\mu$m, the nucleus of NGC 5278 (NGC 5279)
contributes no more than a fraction 0.034 (0.067) of the total system flux
after correction for stellar photospheric light, while at 24$\mu$m, the limit
is 0.31 (0.23). If these values are representative of the system at longer
wavelengths, the ratios in Table 4 should be divided by these fractions, which
would push the NGC 5278/9 EELR clouds into the regime requiring a fading AGN.
Clearly this question will benefit from improved angular resolution throughout
the infrared. An order-of-magnitude improvement in discrimination of AGN from
surrounding star formation at these redshifts, for example, would come from
quite short exposures using MIRI on JWST.

Similarly, X-ray measurements can isolate AGN emission and provide strong
constraints on the ionizing continuum, with appropriate caution about the
possibility of the source being strongly obscured at energies below $\approx
5keV$ by surrounding material at high column densities. For the EELR hosts in
the TELPERION samples, the data do not yet exist to use this test at any
greater sensitivity than spatially integrated far-IR fluxes. The bright AGN in
this group, NGC 235, Mkn 266, and Mkn 1, which are energetically capable of
photoionizing the EELR according to the simple model described above, have
X-ray detections that are compatible with this conclusion. NGC 7252, with no
optical or IR evidence of an AGN, has an X-ray luminosity and spectral
parameters compatible with emission from its young stellar population and ISM
(Brightman & Nandra, 2011). All the other TELPERION EELR hosts - NGC 5278/9,
NGC 5514, and UGC 5941 - still lack even catalog X-ray detections which are
usefully deep for this purpose. Scaling from the bright detections suggests
that point-source sensitivity near $5\times 10^{-14}$ erg cm-2 s-1 from 2-10
keV would show more conclusively whether, for example, all the clouds in the
NGC 5278/9 system show fading of the AGN, anchoring the SED of each AGN at
energies where other sources are weaker and can be estimated from the stellar
population of the host galaxy.

Table 4: Cloud-nucleus energy balance data Object | $z$ | Type | L${}_{\rm[O~{}III]}$, erg s-1 | LBol | Lion,cloud | Ratio | Projected distance, kpc
---|---|---|---|---|---|---|---
Mkn 1 | 0.016 | Sy 2 | $3.0\times 10^{41}$ | $5.1\times 10^{44}$ | $6\times 10^{42}$ | <34 | 13
NGC 7252 | 0.016 | — | - | - | — |  | 13
NGC 235 | 0.022 | Sy 2 | $1.3\times 10^{41}$ | $2.8\times 10^{44}$ | $2.4\times 10^{44}$ | <1.16 | 26
NGC 5514 | 0.024. | Sy 2 | $1.1\times 10^{39}$ | $1.4\times 10^{44}$ | $3.9\times 10^{44}$ | <0.35 | 75
Mkn 266 | 0.028 | Sy2x2 | $6.2\times 10^{40}$ | $1.1\times 10^{45}$ | $3.9\times 10^{44}$ | <2.8 | 25
NGC 5278 inner | 0.025 | LINER/Sy 2 | $6.8\times 10^{39}$ | $8.0\times 10^{43}$ | $3.2\times 10^{43}$ | <2.5 | 14
NGC 5278 outer | 0.025 | LINER/Sy 2 | $6.8\times 10^{39}$ | $8.0\times 10^{43}$ | $2.1\times 10^{44}$ | <0.38 | 29
NGC 5279 | 0.025 | LINER/Sy 2 | $1.0\times 10^{40}$ | $8.1\times 10^{43}$ | $2.9\times 10^{44}$ | <0.28 | 18
UGC 5941 | 0.024 | Star-forming |  $1.2\times 10^{40}$FAGN | $2.7\times 10^{44}$ | $4.7\times 10^{44}$ | <0.6 | 35
Figure 5: Comparison of the TELPERION EELRs, to the same scale in linear size
and (approximately) the same [O III] surface-brightness scale. Mottling in the
host-galaxy region of UGC 5941 is a result of imperfect matching of non-
Gaussian PSFs. NGC 7252 is shown in data from the 2.5m DuPont telescope at Las
Campanas, from Schweizer et al. 2013. Data for Mkn 1 and NGC 235 are from SARA
images. Mkn 1 shows artifacts due to imperfect removal of reflections from a
bright star, but the ionization cones identified by Stoklasová et al. 2009
appear on either side of the nucleus. For Mkn 266, we show the intensity
derived from fits through a Fabry-Perot data cube obtained with the SCORPIO-2
system at the BTA. The NGC 5514 and 5278/9 data used the tunable-filter MaNGaL
system at the Caucasus Mountain Observatory 2.5m telescope (section 2.4). Each
image is rotated for convenient display. Small EELRs are indicated by tick
marks above and below. For three systems with multiple nuclei, their locations
are shown by white circles.

##  4 Conclusions

We present the results of a narrow-band imaging search for distant EELRs
(extending $>10$ kpc from the nucleus) around galaxies with and without AGN
seen directly, to assess the luminosity history of the AGN as well as
contribute to our understanding of the pattern of emerging radiation from the
cores. In a sample of 198 strongly-interacting and merging galaxies within 92
systems, we have identified 3 previously-identified and one newly recognized
distant EELR, with an AGN too weak to separate from signatures of ongoing star
formation but with the cloud lying in the AGN-ionized regions of BPT
classification diagrams. Collecting these results with the three previous
TELPERION samples, we find detection rates of distant EELRs from 20-40 times
higher than in the Galaxy Zoo survey, which examined a much larger galaxy
sample to [O III] surface brightness levels several times higher than reached
with our current narrow-band images. Detection rates for distant EELRs around
interacting Seyfert galaxies are particularly high, $12\pm 6$%, continuing the
earlier preponderance of EELRs in interacting and merging systems. This must
at least in part reflect the presence of distant, often extraplanar H I in
tidally distorted galaxies.

One of our motivations for the TELPERION survey was to assess the duration of
bright periods in episodic AGN accretion. We have detected EELRs around AGN
sufficiently faint that the simple picture used earlier to ask whether AGN are
more likely to be obscured or faded can no longer be applied with available
data, because the integrated far-IR output is dominated by surrounding star
formation and X-ray catalogs are not yet deep enough. Analysis of this kind
will need to await higher-resolution data in the mid and far-infrared or
deeper high-resolution X-ray observations.

## Acknowledgements

The narrowband filters were obtained thanks to a University of Alabama College
of Arts and Sciences Dean’s Leadership Board Faculty Fellowship. Acquisition
of new imagers for the SARA Observatory was supported by the National Science
Foundation through grant 0922981 to East Tennessee State University, and
retrofitting of the Jacobus Kapteyn Telescope through grant 1337566 to Texas
A&M University – Commerce. We thank Steve Willner and Matt Ashby for copies of
the processed SFRS Spitzer observations, and Francois Schweizer for the Las
Campanas [O III] image of NGC 7252.

Observations with the SAO RAS telescopes are supported by the Ministry of
Science and Higher Education of the Russian Federation. The renovation of
telescope equipment is currently provided within the national project “Science
and Universities". We obtained part of the observed data on the unique
scientific facility “Big Telescope Alt-azimuthal" as well as analysed the
2.5-m and 6-m telescopes data with the financial support of grant
No075-15-2022-262 (13.MNPMU.21.0003) of the Ministry of Science and Higher
Education of the Russian Federation.

The authors are honored to be permitted to conduct astronomical research on
Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the
Tohono O’odham Nation. This research has made use of the NASA/IPAC
Extragalactic Database (NED), which is operated by the Jet Propulsion
Laboratory, Caltech, under contract with the National Aeronautics and Space
Administration. This research has made use of the VizieR catalogue access
tool, CDS, Strasbourg, France (DOI : 10.26093/cds/vizier). The original
description of the VizieR service was published in 2000, A&AS 143, 23.

Funding for the Sloan Digital Sky Survey V has been provided by the Alfred P.
Sloan Foundation, the Heising-Simons Foundation, the National Science
Foundation, and the Participating Institutions. SDSS acknowledges support and
resources from the Center for High-Performance Computing at the University of
Utah. SDSS telescopes are located at Apache Point Observatory, funded by the
Astrophysical Research Consortium and operated by New Mexico State University,
and at Las Campanas Observatory, operated by the Carnegie Institution for
Science. The SDSS web site is www.sdss.org.

SDSS is managed by the Astrophysical Research Consortium for the Participating
Institutions of the SDSS Collaboration, including Caltech, The Carnegie
Institution for Science, Chilean National Time Allocation Committee (CNTAC)
ratified researchers, The Flatiron Institute, the Gotham Participation Group,
Harvard University, Heidelberg University, The Johns Hopkins University,
L’Ecole polytechnique fédérale de Lausanne (EPFL), Leibniz-Institut für
Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA
Heidelberg), Max-Planck-Institut für Extraterrestrische Physik (MPE), Nanjing
University, National Astronomical Observatories of China (NAOC), New Mexico
State University, The Ohio State University, Pennsylvania State University,
Smithsonian Astrophysical Observatory, Space Telescope Science Institute
(STScI), the Stellar Astrophysics Participation Group, Universidad Nacional
Autónoma de México, University of Arizona, University of Colorado Boulder,
University of Illinois at Urbana-Champaign, University of Toronto, University
of Utah, University of Virginia, Yale University, and Yunnan University.

## Data Availability

The SARA images, and BTA long-slit spectrum of UGC 5941, are available at
Zenodo.org under DOI 10.5281/zenodo.8374640. The Spitzer data may be retrieved
from the Spitzer Heritage Archive at IRSA
(https://irsa.ipac.caltech.edu/applications/Spitzer/SHA).

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