Questions tagged [measurable-functions]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
15 votes
3 answers
3k views

Are measurable homomorphisms $ (\Bbb{C},+) \to (\Bbb{C},+) $ or $ (\Bbb{C},+) \to (\Bbb{C},*) $ continuous, and do they admit an explicit description?

I am interested in generalizations of the following fact (known as automatic continuity, as pointed out below). I am especially looking for references to papers dating back to 1920’s. I feel that ...
mmm 's user avatar
  • 1,299
11 votes
2 answers
10k views

The image of a measurable set under a measurable function.

Let $f:X \rightarrow (Y, \mathcal{Y})$ be an abstract function, with $\mathcal{Y}$ a $\sigma$-algebra on $Y$. Endow $X$ with $f^{-1}(\mathcal{Y})$. Is then $f(X)$ a measurable set in $Y$? If not, are ...
Alex M.'s user avatar
  • 5,207
11 votes
2 answers
3k views

Good examples of random variables whose image is not a measurable set?

Are their simple/natural examples of real-valued Borel-measurable random variables whose image is not a Borel set? Something that occurs "naturally"? I am teaching Doob's lemma (for two real-valued ...
Uwe Franz's user avatar
  • 2,191
10 votes
1 answer
1k views

When is the set of measurable functions a vector space?

I know this is not a research question, but I searched somewhat thoroughly and could not find the exact answer I want. But I've always wondered the following: suppose that $(X,\mathcal{M})$ is a ...
UserA's user avatar
  • 587
10 votes
1 answer
1k views

Is the "continuous on compact subsets" characterization of measurable functions actually useful?

According to Lusin's theorem (and the slightly weaker converse of that result), measurable functions on locally compact topological spaces that are equipped with a regular measure may be characterized ...
user avatar
9 votes
2 answers
642 views

Analogue of open/closed maps for measurable spaces

$\newcommand{\A}{\mathcal{A}}\newcommand{\T}{\mathcal{T}}$The notions of continuous map of topological spaces and measurable function of measurable spaces are very similar: A map of topological ...
Emily's user avatar
  • 10.3k
6 votes
1 answer
1k views

Are Bochner measurablity and Borel measurability compatible?

Let $(X,\mathfrak{M})$ be a measurable space and $E$ be a Banach space and $f:(X,\mathfrak{M})\rightarrow E$ be a function. Question Are Borel-measurable condition on $f$ and Bochner-measurable ...
Rubertos's user avatar
  • 337
6 votes
0 answers
335 views

Radon-Nikodym derivatives with parameters?

Suppose that $(A,\Sigma_A)$ and $(X,\Sigma_X)$ are measurable spaces, and that $$ \mu,\nu \: : \: A \times \Sigma_X \rightarrow [0,1] $$ are Markov kernels, i.e. probability measures on $X$ ...
Tobias Fritz's user avatar
  • 5,775
6 votes
0 answers
357 views

Approximating a measurable function from a second-countable, locally compact Hausdorff group to a separable Banach space

Let $ G $ be a second-countable, locally compact Hausdorff group and $ B $ a separable Banach space. We say that a function $ f: G \to B $ is Bochner-measurable if and only if it is the everywhere ...
Transcendental's user avatar
5 votes
1 answer
292 views

Continuity of real functions

The following question concerns that without $ZF+DC$, can every function be "a little bit" continuous? Question Is it consistent with $ZF+DC$ that for any function $f:[0,1]\to [0,1]$ and ...
喻 良's user avatar
  • 4,191
5 votes
1 answer
368 views

Is there a Borel measurable $f:\mathbb{R}^d \to \mathbb{R}^d$ such that $f(x) \in \partial \varphi (x)$ for all $x$?

Let $\varphi: \mathbb{R}^d \to \mathbb{R}$ be a convex function. The subdifferential of $f$ at $x$ is defined as $$ \partial \varphi (x) := \{z \in \mathbb{R}^d : \varphi(y) \geq \varphi(x) + \langle ...
Akira's user avatar
  • 815
5 votes
2 answers
628 views

Exotic Lebesgue Measurable Function

Measurable functions whose graphs are dense in the plane are well known. Examples include, the Conway 13 function, as given in the answer in this link: When is the graph of a function a dense set? ...
topsyturvy's user avatar
5 votes
1 answer
202 views

Dense subcategory of measurable spaces

Recall the notion of a dense subcategory $\mathcal{D}$ of a category $\mathcal{C}$. It means that the restricted Yoneda functor $\mathcal{C} \to \mathrm{Hom}(\mathcal{D}^{op},\mathbf{Set})$, $A \...
Martin Brandenburg's user avatar
4 votes
1 answer
200 views

Measurable selection involving measure valued random variable

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{M}(\mathbb{R}^d)$ be the space of finite signed measures on $\mathbb{R}^d$ endowed with the narrow topology (i.e. the ...
Bremen000's user avatar
  • 327
4 votes
1 answer
735 views

The Notion of Strong Measurability for Separable Banach Spaces

Let $ (X,\Sigma,\mu) $ be a measure space and $ B $ a Banach space. According to my understanding, a function $ f: X \to B $ is said to be strongly $ \mu $-measurable if and only if it is the almost-...
Transcendental's user avatar
4 votes
1 answer
261 views

Volume-preserving mappings in the torus $T^n$

Let $T^n$ be the $n$-dimensional torus and let $F$ be the set of all volume preserving continuous mappings $f:T^n\to T^n$. I would like to know if $F$ is connected in the sense that for any $f\in F$ ...
John Habitts's user avatar
4 votes
1 answer
520 views

Optimal Transport: how is this transport map Borel measurable?

I'm reading Theorem 1.17. and its proof at page 14 of Santambrogio's Optimal transport for applied mathematicians. The content is not hard but a little bit long (because of related detail). Please ...
Akira's user avatar
  • 815
3 votes
2 answers
903 views

Opinions on the Multiplication of Measures

A few questions, hopefully to spark some discussion. How can one define a product of measures? We could use Colombeau products by embedding the measures into the distributions? I'm not sure why this ...
dcs24's user avatar
  • 213
3 votes
1 answer
139 views

Measurability of superposition operator with non-separable Banach space

Let $f\colon I \times X \to \mathbb{R}$ be a map where $I \subset \mathbb{R}$ is an interval, $X$ is a Banach space (possibly non-separable) and we have $$t \mapsto f(t,x) \text{ is measurable}$$ $$x \...
MMML's user avatar
  • 107
3 votes
1 answer
220 views

Measurable selection for argmin to distance

Let $Y$ be a Banach space and equip $Y$ with the weak topology. Now, let $X$ be a closed, bounded, and convex subset of $Y$ and suppose that the relative (weak) topology on $X$ is metrizable with ...
ABIM's user avatar
  • 5,019
3 votes
2 answers
405 views

Sharp assumption for preserving Lebesgue measurability by left composition

Let $g: [0, 1] \to \mathbb R$ be a Lebesgue-measurable function (in the classical sense: the inverse images of Borel sets are Lebesgue-measurable). It is a classical fact in analysis that $f \circ g$ ...
Guilherme Mazanti's user avatar
3 votes
3 answers
750 views

When Banach indicatrix is measurable?

Let $f:X\to Y$ is a measurable function. Banach indicatrix $$ N(y,f) = \#\{x\in X \mid f(x) = y\} $$ is the number of the pre-images of $y$ under $f$. If there are infinitely many pre-images then $N(y,...
Nikita Evseev's user avatar
3 votes
1 answer
145 views

Can there be an upper bound on the Borel rank of the preimages of Borel sets under a surjective Borel map?

Let $X$ and $Y$ be standard Borel spaces, $Y$ uncountable, and $f : X \to Y$ a surjective Borel map. Is it possible that there is a countable ordinal $\alpha$ such that for each Borel set $B \subseteq ...
Arkadi Predtetchinski's user avatar
3 votes
1 answer
283 views

Loeb measures and non-standard hull of Banach spaces

$\DeclareMathOperator\Fin{Fin}$I am trying to understand the notion of "liftings" of Loeb measurable functions to internal, internally measurable functions, and its connection to non-...
BharatRam's user avatar
  • 939
3 votes
1 answer
152 views

$L_p(I,Y)^\perp=L_q(I,Y^\perp)$?

Let $X$ be a Banach space and $Y$ be a closed subspace of $X$. For $1<p<\infty$ consider the $p$-th power Bochner Integrable functions which takes values in $X$ and defined on the unit interval $...
Tanmoy Paul's user avatar
3 votes
0 answers
176 views

Example of an optional non-predictable process

To clarify better the notions of predictable and optional processes, I am looking for a simple example of a process that is optional, but not predictable. I found out something useful here, however, I ...
AlmostSureUser's user avatar
3 votes
0 answers
203 views

Characterization of a Bochner/strongly measurable function solely as a random element

Be $(\Omega, \mathcal{A}, P)$ and $E$ a probability space and a Banach space respectively. This paper of G.A. Edgar contains a proof that, for a function $X: \Omega \rightarrow E$, being weakly ...
M1011's user avatar
  • 31
3 votes
1 answer
222 views

Sobolev embedding into measurable functions

Consider the fractional Sobolev space $$ W^{k,2}(\mathbb R^n):=\big\{f \in \mathcal S'\,\big|\,(1+\|\xi\|)^k\hat f(\xi)\in L^2(\mathbb R^n)\big\} $$ for some $k\in\mathbb R$, and let $\mathcal M$ ...
André Henriques's user avatar
3 votes
0 answers
110 views

A measurable implicit function with differentiated arguments

I have encountered the need for an unusual implicit function theorem, about which I know very little. I would appreciate it if someone could help me with a few pointers. The setup is as follows. Let $...
Ankur's user avatar
  • 183
2 votes
1 answer
246 views

Is scalarly measurable simply measurable?

More specifically, consider the following particular situation: Let $I=[0,1]$ with the standard Borel $\sigma$-algebra. Consider functions $y:I\times I\to I$. Say that $y$ is scalarly measurable iff $...
TaQ's user avatar
  • 3,338
2 votes
2 answers
226 views

Preimage of null sets under a monotone increasing function

Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...
Julian's user avatar
  • 113
2 votes
1 answer
716 views

Functional representation of adapted jointly measurable stochastic processes

It seems like the question stated here in MSE has no answer yet and seems therefore for me to be not of a basic question type. For this reason I move it to MO. Let $X_t : \Omega \to E, \ t \geq 0$ be ...
yada's user avatar
  • 1,731
2 votes
1 answer
298 views

Measurability of Markov kernel wrt the Borel $\sigma$-algebra generated by the weak topology

Consider two Polish metric probability spaces $(\mathcal{A}, \Sigma_\mathcal{A})$ and $(\mathcal{B}, \Sigma_\mathcal{B})$, endowed with their Borel $\sigma$-algebras. Denote as $\mathcal{P}_\mathcal{B}...
ECL's user avatar
  • 271
2 votes
1 answer
199 views

Measurability of maximum likelihood estimator under conditions from Lehmann's "Theory of point estimation"

I'm trying to prove that MLE from the proof of one theorem in Lehmann's "Theory of point estimation" (the theorem is below) is a measurable function. I know that under some regularity ...
Botnakov N.'s user avatar
2 votes
2 answers
2k views

Supremum of continuous functions and essential supremum of continuous functions

Suppose that $(X,d)$ is a Polish metric space and $A$ is a set of continuous bounded functions $f:X\to \mathbb{R}$. Suppose that $\mu:X\to[0,1]$ is a Borel probability measure. Define $$\sup A:X\to ...
Littlefield's user avatar
2 votes
1 answer
421 views

Is the following "section-wise" defined function measurable in the product space?

I asked this question in mathstackexchange a couple of days ago. Almost right after posing it a partial (affirmative) answer came to my mind in the following form Proposition: Assume that $(X,\...
David's user avatar
  • 486
2 votes
0 answers
60 views

On the measurability of stochastic integrals

Let $S(t)$ be a $C_0$-contraction semi-group, $W$ is a cylindrical Wiener process in a separate Hilbert space $U$. Assume the following conditions: $$ \|F(t,u_1)-F(t,u_2)\|_{H}< C\|u_1-u_2\|_{H},~~...
ABCD's user avatar
  • 31
2 votes
0 answers
62 views

Measure of subsets in $\mathbb S^d$ defined by multiplicities of real roots

We associate to an element $\mathbf x=(x_0,\ldots,x_d)$ of the real unit sphere $\mathbb S^d=\lbrace (x_0,\ldots,x_d)\in\mathbb R^{d+1},\ x_0^2+\dots+x_d^2=1\rbrace$ the number $\mu(\mathbf x)$ of ...
Roland Bacher's user avatar
2 votes
0 answers
52 views

The initial sigma-algebra on the dual of a Banach lattice

Let $E$ be an AL space (i.e. a Banach lattice whose norm is additive on the positive cone $E_+$) that satisfies Mazur's condition (every sequentially weak$^*$-continuous functional on $E'$ is weak$^*$-...
HardyHulley's user avatar
2 votes
0 answers
201 views

Prove or disprove that $u=0$ a.e. on $\Bbb R^d$

Let $\Omega\subset\Bbb R^d$ be an open set. Let $k:\Bbb R^d\to [0,\infty)$ be measurable such that $0\in \operatorname{supp}k$. This implies that $\Omega\subset \Omega_k=\Omega+\operatorname{supp}k$. ...
Guy Fsone's user avatar
  • 1,033
2 votes
0 answers
445 views

Does equality almost everywhere on a product imply equality almost everywhere on sections [closed]

(This question was on MSE, with no answers) Consider two $\sigma$-finite measure spaces $X_1$ and $X_2$, and $X=X_1\times X_2$ the product measure space (a priori non-completed). Take two functions ...
Jon-S's user avatar
  • 539
2 votes
0 answers
188 views

Progressive measurability and functional composition

Suppose we have a progressively measurable process $X$ taking values in $\mathbb{R}^d$. What are sufficient conditions on a function $$f( x, t, \omega ) \colon \mathbb{R}^d \times [0,\infty) \...
Tobi's user avatar
  • 21
2 votes
1 answer
172 views

Measurability of functions with multiple parameters

For a formalisation of the Giry monad in a theorem prover, I think I require some notion of measurability of “curried” functions. I.e. I have measure spaces $A$, $B$, and $C$ and a function $f: A \...
Manuel Eberl's user avatar
  • 1,181
1 vote
2 answers
173 views

Measurable selection for maximum process

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ be a complete filtered probability space, where $(\mathcal{F}_t)_{t\geq 0}$ is the completed Brownian filtrate. Suppose that $\Phi(t,x)$ ...
Matt Rosenzweig's user avatar
1 vote
1 answer
94 views

Implications of a Regularity Condition for Functions

$ \newcommand{\essSup}{\mathop{\rm sup_{ess}}\nolimits} $ What can be concluded from the fact, that $f: X\ni x\mapsto f(x)\in [a,b]\subset\mathbb{R}\setminus\lbrace{-\infty,+\infty\rbrace} $ ...
Manfred Weis's user avatar
  • 12.6k
1 vote
2 answers
176 views

Does a measurable $F :[0, T] \to L^p (\mathbb R^d; \mathbb R_{\ge 0})$ have a "flattened" measurable version?

Let $d \in \mathbb N^*,p \in [1, \infty]$ and $T>0$. Let $$ F :[0, T] \to L^p (\mathbb R^d; \mathbb R_{\ge 0}), t \mapsto F_t $$ be measurable. I would like to ask if there is a measurable function ...
Akira's user avatar
  • 815
1 vote
1 answer
55 views

Let $c: X \times Y \to \overline{\mathbb R}$ be $\gamma$-measurable. Is $c_x:Y \to \overline{\mathbb R}, y \mapsto c(x, y)$ $\nu$-measurable?

Let $(X, \mathcal X, \mu)$ and $(Y, \mathcal Y, \nu)$ be $\sigma$-finite measure spaces. Let $\overline{\mathbb R} := \mathbb R \cup \{\pm \infty\}$. $f:X \to \overline{\mathbb R}$ is called $\mu$-...
Akira's user avatar
  • 815
1 vote
1 answer
181 views

Friedland metric entropy

I was asking if it is possible to extend the definition of topological Friedland entropy for $\mathbb{Z}^d$ continuos actions to measure preserving actions. The topologica Friedland entropy is ...
user502940's user avatar
1 vote
0 answers
85 views

$f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?

Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...
vaoy's user avatar
  • 287
1 vote
0 answers
46 views

Is $L^p_\text{loc} (Y)$ dense in $(L^0(Y), \hat \rho)$?

Below we use Bochner measurability and Bochner integral. Let $(Y, d)$ be a separable metric space, $\mathcal B$ Borel $\sigma$-algebra of $Y$, $\nu$ a $\sigma$-finite Borel measure on $Y$, $(Y, \...
Analyst's user avatar
  • 647