Questions tagged [measurable-functions]

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2
votes
1answer
114 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 $\mathcal M$ ...
-3
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0answers
33 views

Convergence in distribution (measurable functions)

The definition of convergence in distribution: Let $S$ be a seperable metric space and $(X_n)_{n\in\mathbb{N}}$ a $S$-valued sequence of random variables. Then $(X_n)_{n\in\mathbb{N}}$ converge in ...
2
votes
1answer
85 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 ...
5
votes
1answer
279 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 ...
1
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0answers
81 views

$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)} \mathrm{d} m=0 $ associated with convergence in measure [closed]

For $m E<+\infty$, why the sufficient and necessary condition of $\left\{f_{n}(x)\right\}$ converge in measure to $0$ is $$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)}...
0
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0answers
78 views

Measurability of centered Hardy-Littlewood maximal function with doubling measure

Let $(X, d, \mu)$ be a metric space with doubling measure $\mu$ a.e. every open ball has finite and positive measure $\mu$ and there exists $C>0$ such that $$\mu(B(x,2r)) \le C\mu(B(x,r))$$ for ...
5
votes
1answer
174 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 \...
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0answers
99 views

What are the functions such that $ \lVert f + g \rVert_p^p = \lVert f \rVert_p^p + \lVert g \rVert_p^p$?

Let $1 \leq p \leq 2$. I am looking for a characterization of the couples $(f,g)$ of functions $f,g \in L_p(\mathbb{R})$ such that $$ \lVert f + g \rVert_p^p = \lVert f \rVert_p^p + \lVert g \rVert_p^...
0
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1answer
144 views

$f_\epsilon=\inf\{f(y):|y-x|<\epsilon\}$ is measurable Borel [closed]

I found this problem I have tried but it has been a bit complicated for me, Let $f:\mathbb{R}\to\mathbb{R}$ a bounded function. For each $\epsilon>0$, let $f_\epsilon (x)=\inf\{f(y):|y-x|<\...
3
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1answer
126 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 ...
3
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1answer
126 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 ...
0
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0answers
52 views

Sequence of open sets converge in characteristic function to an open set?

Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set with Lipschitz boundary. Consider a sequence of open sets $\omega_n\subseteq\Omega,\ n\in\mathbb{N}^*$ such that there is a Lebesgue ...
0
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1answer
44 views

Countable sup property of extended measurable functions

Let $(S,\Sigma,\mu)$ a $\mu$-finite measure space. Denote by $\bar{L}^0(\Sigma)$ the set of extended-real valued $\Sigma$-measurable functions. Does this set have the countable sup property when ...
8
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1answer
539 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 ...
3
votes
1answer
183 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-...
3
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1answer
91 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 \...
3
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1answer
105 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 ...
3
votes
2answers
176 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 $f \circ g$ ...
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0answers
78 views

Measurability of infimum function

In Theorem 14.37 of Variational Analysis by Rockafellar and Wets, it shows that for any normal integrand $f:T\times \mathbb{R}\to\mathbb{R}$, the function $p:T\to \mathbb{R}$ given by $p(t):=\inf f(t,⋅...
0
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1answer
119 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 ...
3
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1answer
142 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 $...
1
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1answer
347 views

Transport of measure

Let's disintegrate $\mu$ and $\nu$, two probabilities on $\mathbb{R}^{d}$ , according to $$ \pi_{k} (x_{1},...,x_{d}) = (x_{k},...,x_{d}) $$ We get a family of measures and each measure $\mu_{k,d}^{+...
2
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2answers
821 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 ...
5
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0answers
184 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 $X$ ...
1
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2answers
138 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 $\Phi(t,x)$ ...
0
votes
2answers
108 views

dense subalgebra in measurable functions set

Take $\mathcal M_D$ the space of measurable functions from a compact set $D\subseteq \mathbb R_n$ to ℂ. I'm wondering if a Stone-Weierstrass-like theorem holds in this space, with the convergence in ...
4
votes
1answer
693 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 ...
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0answers
114 views

Is total variation $\mu(\cdot) \mapsto |\mu|(\cdot)$ Borel measurable from $M$ to $M$?

Let $M$ be the space of finite signed measures on $\mathbb{R}$, equipped with the topology of weak convergence of measures. I would like to know if taking the total variation of a measure is a ...
1
vote
1answer
78 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} $ ...
2
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0answers
257 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 ...
3
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0answers
107 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 $...
0
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2answers
161 views

Weak continuity of a vector valued function [closed]

Let $f:[0,1]\to \ell_\infty[0,1]$ be defined by $f(t)=\chi_{[0,t]}$. Is it true that $f$ is weakly continuous almost everywhere w.r.t. Lebesgue measure ? Here $\ell_\infty[0,1]$ represents the ...
3
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3answers
580 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,...
2
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1answer
326 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,\...
2
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0answers
159 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) \...
5
votes
2answers
560 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? ...
1
vote
1answer
595 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 ...
6
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0answers
334 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 ...
4
votes
1answer
418 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-...
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0answers
195 views

Measurability of a map that takes a functional to its composition with a linear operator

Let $(X,\Sigma_X)$ be a measurable space such that $\Sigma_X$ is countably generated. Let $B_b(X)$ be the Banach space of all bounded $\Sigma_X$-measurable functions $X\to\mathbb{R}$ equipped with the ...
2
votes
1answer
154 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 \...
10
votes
1answer
981 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 ...
4
votes
1answer
247 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 $f\in F$ ...
1
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0answers
280 views

Is reflexive Banach space valued scalarwise Lebesgue space isomorphic to the Bochner space?

I first specify the setting and then formulate the question precisely. (A very long post follows.) Definitions 1. For $E$ a (real Hausdorff) locally convex space, say that $E$ is suitable iff there ...
2
votes
1answer
227 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 $...
6
votes
2answers
7k 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 ...
1
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0answers
122 views

Is scalarwise measurability determined by the strong dual?

Since this question has not received an answer so far, I try to reformulate the question in a simpler manner as follows: Do there exist $E,F,\ell,f$ such that $E$ and $F$ are separable (real) Banach ...
10
votes
2answers
2k 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 ...
2
votes
2answers
725 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 ...
1
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0answers
353 views

Unambiguous "weak" vector valued $L^{+\infty}$ spaces?

For some time, I have been stuck to the problem to be described as follows. The (perhaps not so commonly known) facts given here are taken from R. E. Edwards' Functional Analysis (Holt, Rinehart and ...