Questions tagged [measurable-functions]

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2 votes
0 answers
46 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$^*$-...
2 votes
0 answers
195 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 $\Omega\subset \Omega_k=\Omega+\operatorname{supp}k$. ...
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1 vote
0 answers
34 views

Let $(X, W)$ be a weak solution to a SDE. Is $W$ a Brownian motion w.r.t. $\sigma(X_s : s \le t)$?

Let $(X, W)$, $(\Omega, \mathcal{F}, \mathbb{P})$, $\{\mathcal{F}_t\}$ be a weak solution to an SDE. Per definition $W$ is an $\mathcal{F}_t$-Brownian motion and both $X$, $W$ are adapted to $\mathcal{...
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3 votes
0 answers
129 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 ...
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0 votes
1 answer
134 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}...
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3 votes
1 answer
184 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 $\mathcal M$ ...
2 votes
1 answer
122 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
1 answer
284 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 ...
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1 vote
0 answers
85 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)}...
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0 votes
0 answers
175 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 ...
  • 105
5 votes
1 answer
185 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 \...
0 votes
0 answers
100 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^...
  • 2,082
0 votes
1 answer
159 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|<\...
4 votes
1 answer
151 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 ...
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3 votes
1 answer
135 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 votes
0 answers
59 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 ...
  • 1,092
0 votes
1 answer
58 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 votes
1 answer
849 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 ...
  • 537
3 votes
1 answer
245 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-...
  • 929
3 votes
1 answer
108 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 \...
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3 votes
1 answer
147 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 ...
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3 votes
2 answers
289 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 $f \circ g$ ...
0 votes
0 answers
87 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,⋅...
1 vote
1 answer
168 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 votes
1 answer
145 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 vote
1 answer
490 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}^{+...
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2 votes
2 answers
1k 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 ...
6 votes
0 answers
214 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 $X$ ...
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1 vote
2 answers
145 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
2 answers
116 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 ...
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4 votes
1 answer
857 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 ...
  • 307
1 vote
0 answers
119 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 ...
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1 vote
1 answer
87 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} $ ...
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2 votes
0 answers
344 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 ...
  • 529
3 votes
0 answers
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 $...
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0 votes
2 answers
171 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 votes
3 answers
655 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 votes
1 answer
373 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,\...
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2 votes
0 answers
172 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) \...
  • 21
5 votes
2 answers
601 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
1 answer
654 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 ...
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6 votes
0 answers
348 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
1 answer
570 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-...
1 vote
0 answers
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 ...
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2 votes
1 answer
160 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 \...
  • 1,159
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
4 votes
1 answer
253 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 vote
0 answers
284 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 ...
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2 votes
1 answer
231 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 $...
  • 2,968
8 votes
2 answers
8k 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 ...
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