Questions tagged [measurable-functions]
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93 questions
2 votes
0 answers
90 views
Nemytskii operator on $L^2$ space
Let $(\Omega,\mathcal{F},\mu)$ be a measure space and consider a function $f \colon \mathbb{R} \times \Omega \to \mathbb{R}.$ For the problem I work on, a seemingly good hypothesis to place on $f$ is ...
3 votes
0 answers
225 views
Measurable selection of subdifferential of convex function
Let $(X, \Sigma)$ be a measurable space. $ f: X \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ is a $\Sigma \otimes \mathcal{B}\left(\mathbb{R}^{n}\right)$-measurable function and $\forall x \in X, y \...
- 93
0 votes
0 answers
432 views
Performing an uppersemicontinuous regularization twice
I am unsure about the conclusion of a proof I wrote, the statement I am trying to prove as well as the proof itself are included below. Exact statement I am trying to prove: Let $f(\lambda,z): \mathbb{...
- 61
2 votes
1 answer
208 views
Conditional expectation of "multi-stage" experiment
Suppose $X_t$ is a collection of random variables from a measure space $(\Omega, \mathcal{F}, P)$ to $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ (to keep things simple), where $t \in \mathbb{R}$. Moreover ...
1 vote
1 answer
147 views
Is the map $\Omega \ni \omega \mapsto \partial_x f (\omega, x) \in X^*$ measurable?
$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bP}{\mathbb{P}} \newcommand{\cA}{\mathcal{A}} $ Let $(\Omega, \cA, \bP)$ be a complete probability space. Let $(X, \| \cdot \|)$ be a real separable Banach ...
- 1,163
4 votes
1 answer
169 views
Covering with small measure a subsequence of a sequence of small-measure sets
I self-study measure theory from Stein-Shakarchi. For practice, I tried to prove on my own that $L^1$ is complete (without having seen the proof). During my attempt, I got stuck on the following ...
- 464
4 votes
1 answer
276 views
Problem in Probability Theory and Functional Analysis
Let's consider the vector space V of bounded scalar functions, which includes the constant function 1. We assume that any uniform limit of a bounded monotonic sequence of functions from V also ...
- 51
1 vote
0 answers
289 views
If $f \in L^p(\Omega)$, then $f \in L^q(\Omega)$ for some $q < p + \epsilon$?
Loosely speaking, I would like to know whether membership in some Lebesgue space $L^p$ is stable under small perturbations of the exponent $p$. Let $\Omega \subseteq \mathbb R^n$ be a bounded domain ...
- 641
7 votes
1 answer
298 views
Can we generalize the Kuratowski Extension Theorem to Souslin spaces?
The Kuratowski Extension Theorem says: Let $(X,\mathcal{A})$ be a measurable space, $Y$ be a polish space, $A\subseteq X$, and $f:A\to Y$ be a measurable map. Then there is a measurable function $F:X\...
- 763
1 vote
1 answer
269 views
Function $g:\mathbb{R}\to \mathbb{R}^n$ such that $g(\sum_{i=1}^nx_i)=(x_1,\dotsc,x_n)$ a.e
Is there a measurable function $g:\mathbb{R}\to \mathbb{R}^n$ such that $g(\sum_{i=1}^nx_i)=(x_1,\dotsc,x_n)$ a.e.? Due to the papers [1], [2], and [3] I'm obtaining a result that I think it's false. ...
- 763
3 votes
1 answer
288 views
Compactness of set of measurable functions between compact subspaces of real numbers
Let $X$ be a compact subset of $\mathbb{R}^n$ and $Y$ be a convex and compact subset of $\mathbb{R}^p$. Consider $\mathcal{F}$ the set of all measurable functions from $X$ to $Y$. Can I find ...
- 155
1 vote
1 answer
247 views
A question on Borel measurability
Let $(X, \mathcal{B}_{X}, \mu)$ be a measure space. Here, $\mu$ is an infinite Borel measure and $\mu$ is not $\sigma$-finite. Let $\pi$ be surjective Borel measurable map form $(X, \mathcal{B}_{X}, \...
- 11
2 votes
1 answer
104 views
Approximate a non-negative function which is measurable in product $\sigma$-algebra
$ \DeclareMathOperator*{\supp}{supp} \newcommand{\bR}{\mathbb{R}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\...
- 1,163
1 vote
2 answers
173 views
Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integrable. Is $D$ measurable?
$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bD}{\mathbb{...
- 1,163
2 votes
2 answers
277 views
Measurable extension
Let $(\Omega, \mathcal{F})$ be a measurable space and $X$ some metric space (probably Polish) with the Borel $\sigma$-algebra and a function $f: \Omega \times X \to \mathbb{R}$. Usually, functions ...
- 166
4 votes
1 answer
250 views
Product map on topological group measurable?
Let $G$ be a topological group and $\mathcal{B}$ its Baire $\sigma$-algebra (i.e. the smallest $\sigma$-algebra for which all continuous functions $G\rightarrow\mathbb{R}$ are measurable). Consider ...
- 181
0 votes
0 answers
102 views
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 ...
- 1,007
0 votes
0 answers
76 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 ...
2 votes
2 answers
462 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 ...
- 113
1 vote
0 answers
110 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 ...
- 329
-1 votes
1 answer
292 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 ...
- 3,745
0 votes
1 answer
113 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 ...
- 3,745
2 votes
0 answers
89 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: $$ \|F(t,u_1)-F(t,u_2)\|_{H}< C\|u_1-u_2\|_{H},~~...
- 31
6 votes
1 answer
565 views
Is there a Borel measurable $f:\mathbb{R}^d \to \mathbb{R}^d$ such that $f(x) \in \partial \varphi (x)$ for all $x$?
Let $\varphi: \mathbb{R}^d \to \mathbb{R}$ be a convex function. The subdifferential of $f$ at $x$ is defined as $$ \partial \varphi (x) := \{z \in \mathbb{R}^d : \varphi(y) \geq \varphi(x) + \langle ...
- 1,163
1 vote
0 answers
118 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, \...
- 657
0 votes
0 answers
70 views
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, \...
- 657
0 votes
0 answers
141 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 ...
- 657
1 vote
0 answers
131 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 ...
- 657
0 votes
2 answers
164 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 ...
- 1,163
0 votes
2 answers
175 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 ...
- 1,163
1 vote
2 answers
236 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 ...
- 1,163
1 vote
0 answers
193 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 ...
- 39
2 votes
0 answers
77 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 ...
- 18.4k
9 votes
2 answers
776 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 ...
- 12.6k
1 vote
0 answers
58 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, ...
5 votes
1 answer
818 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 ...
- 1,163
3 votes
0 answers
303 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 ...
- 131
1 vote
1 answer
90 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 $\mu$-...
- 1,163
2 votes
0 answers
80 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$^*$-...
- 153
2 votes
0 answers
217 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 $\Omega\subset \Omega_k=\Omega+\operatorname{supp}k$. ...
- 1,155
3 votes
1 answer
277 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{...
- 31
3 votes
0 answers
291 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 ...
- 31
4 votes
1 answer
432 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}...
- 345
3 votes
1 answer
277 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 $\mathcal M$ ...
- 44.7k
2 votes
1 answer
324 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 ...
- 141
5 votes
1 answer
337 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 ...
- 4,636
1 vote
0 answers
117 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 $$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)}...
- 11
0 votes
0 answers
490 views
Measurability of centered Hardy-Littlewood maximal function with doubling measure
Let $(X, d, \mu)$ be a metric space with doubling measure $\mu$ a.e. every open ball has finite and positive measure $\mu$ and there exists $C>0$ such that $$\mu(B(x,2r)) \le C\mu(B(x,r))$$ for ...
- 147
5 votes
1 answer
266 views
Dense subcategory of measurable spaces
Recall the notion of a dense subcategory $\mathcal{D}$ of a category $\mathcal{C}$. It means that the restricted Yoneda functor $\mathcal{C} \to \mathrm{Hom}(\mathcal{D}^{op},\mathbf{Set})$, $A \...
0 votes
0 answers
134 views
What are the functions such that $ \lVert f + g \rVert_p^p = \lVert f \rVert_p^p + \lVert g \rVert_p^p$?
Let $1 \leq p \leq 2$. I am looking for a characterization of the couples $(f,g)$ of functions $f,g \in L_p(\mathbb{R})$ such that $$ \lVert f + g \rVert_p^p = \lVert f \rVert_p^p + \lVert g \rVert_p^...
- 2,602