Questions tagged [measurable-functions]

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Multiplication with dilations of nonzero measurable function is injective

Denote $f_s(x):=f(sx)$ as the dilation of a function $f$. I want to know whether the following statement is true: Suppose $f$ and $g$ are measurable functions on $\mathbb{R}$, and $f$ is not almost ...
Zhang Yuhan's user avatar
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45 views

Existence of derivative of distribution of exponential family?

Suppose $(X, \mathcal{F})$ is a measurable space and $\left\{F_\theta, \theta \in \Theta\right\}$ is a distribution family on $(X, \mathcal{F})$. When $\left\{F_\theta, \theta \in \Theta\right\}$ is ...
Jaimin Shah's user avatar
2 votes
2 answers
233 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
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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
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-1 votes
1 answer
88 views

Pointwise limit of a "net" of measurable functions is measurable? [closed]

Let $(X, \mathcal{A},\mu)$ be a finite measure space with the $\sigma$-algebra $\mathcal{A}$ and the measure $\mu$. Let $B$ be a separable Banach space. Then, it is well-known from a theorem by Pettis ...
Isaac's user avatar
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1 answer
86 views

A nonlinear mapping on $L^2(S^1)$ that commutes with all translation operators is necessarily measurable?

Let $H:= L^2(S^1)$, where $S^1$ is the circle, and $\tau_a : H \to H$ be the translation operator for each $a \in S^1$: \begin{equation} (\tau_a f)(x):= f(x+a) \end{equation} Then, it is clear that ...
Isaac's user avatar
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0 answers
51 views

Characterizing functions that are limits of integrable lower-bounded functions

Let $X$ be a separable Hausdorff topological space, endowed with a positive finite Borel regular measure. Consider those (trivially measurable) functions $f : X \to \mathbb R$ such that their ...
Alex M.'s user avatar
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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
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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
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1 vote
0 answers
47 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
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Is the embedding $i: (L^p_\text{loc} (Y), \| \cdot \|_{L^p_\text{loc}}) \to (L^0(Y), \hat \rho)$ continuous or Borel measurable?

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
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109 views

The set of measurable functions together with convergence in measure is a completely metrizable abelian topological group

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space, $(E, | \cdot |)$ a Banach space, $S (X)$ the space of $\mu$-simple ...
Analyst's user avatar
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Assume $f(x, \cdot) \in L^p_{\text{loc}} (Y)$ for a.e. $x \in X$. Is $F: X \to L^p_{\text{loc}} (Y), x \mapsto f(x, \cdot)$ Bochner measurable?

Below we use Bochner measurability and Bochner integral. Let $T>0$ and $p \in [1, \infty)$. Let $X :=[0, T]$ and $Y:= \mathbb R^d$. Let $L^p_{\text{loc}} (Y)$ be the space of measurable functions $...
Akira's user avatar
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1 vote
0 answers
79 views

Is this metric on the space of $\mu$-measurable functions complete?

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ be a complete finite measure space, $(E, | \cdot |)$ a Banach space, $S (X)$ the space of $\mu$-simple functions ...
Analyst's user avatar
  • 647
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2 answers
84 views

How to construct this sequence that converges a.e. in product measure and that has a very particular form?

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces, $(E, | \cdot |)$ a Banach space, $S (X)$ the ...
Akira's user avatar
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2 answers
120 views

Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?

Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in this thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue ...
Akira's user avatar
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2 answers
178 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
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1 vote
0 answers
138 views

Polish spaces and analytic sets

Can we conclude that an analytic subset $A$ of a Polish space $X$ is also Polish? Let $\mathcal{M}(R^d)$ denotes the family of Borel probability measures on $R^d$ equipped with the Lévy-Prokhorov ...
B-S's user avatar
  • 39
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
9 votes
2 answers
650 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
  • 10.8k
1 vote
0 answers
43 views

Measurability in a product space of a set defined only along its fibers

Consider the probability space $([0,1],\mathcal{B}([0,1]),\lambda)$, where $\mathcal{B}([0,1])$ denotes the Borel $\sigma$-algebra in $[0,1]$ and $\lambda$ is the Lebesgue measure in $[0,1]$. Then, ...
Giuseppe Tenaglia's user avatar
4 votes
1 answer
531 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
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3 votes
0 answers
180 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
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
  • 851
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
1 vote
1 answer
125 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{...
Lochend's user avatar
  • 11
3 votes
0 answers
210 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
2 votes
1 answer
302 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
3 votes
1 answer
223 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
2 votes
1 answer
212 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
5 votes
1 answer
293 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
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1 vote
0 answers
95 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)}...
Ad_M's user avatar
  • 11
0 votes
0 answers
344 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 ...
elsnar's user avatar
  • 127
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
0 votes
0 answers
108 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^...
Goulifet's user avatar
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0 votes
1 answer
317 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|<\...
Zaragosa's user avatar
  • 123
4 votes
1 answer
204 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
3 votes
1 answer
146 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
0 votes
0 answers
81 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 ...
Bogdan's user avatar
  • 1,330
0 votes
1 answer
83 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 ...
Giuliosky's user avatar
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
  • 597
3 votes
1 answer
286 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
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
226 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
  • 4,969
3 votes
2 answers
407 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
0 votes
0 answers
112 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,⋅...
Giuliosky's user avatar
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
3 votes
1 answer
153 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
1 vote
1 answer
641 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}^{+...
CechMS's user avatar
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