# Questions tagged [measurable-functions]

The tag has no usage guidance.

36 questions
Filter by
Sorted by
Tagged with
73 views

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 $$t \mapsto f(t,x) \text{ is measurable}$$ $$x \... 1answer 75 views ### Measurable selection for argmin to distance Let Y be a Banach space and equip Y with the weak topology. Now, let X be a closed, bounded, and convex subset of Y and suppose that the relative (weak) topology on X is metrizable with ... 2answers 80 views ### Sharp assumption for preserving Lebesgue measurability by left composition Let g: [0, 1] \to \mathbb R be a Lebesgue-measurable function (in the classical sense: the inverse images of Borel sets are Lebesgue-measurable). It is a classical fact in analysis that f \circ g ... 0answers 55 views ### Measurability of infimum function In Theorem 14.37 of Variational Analysis by Rockafellar and Wets, it shows that for any normal integrand f:T\times \mathbb{R}\to\mathbb{R}, the function p:T\to \mathbb{R} given by p(t):=\inf f(t,⋅... 1answer 63 views ### Friedland metric entropy I was asking if it is possible to extend the definition of topological Friedland entropy for \mathbb{Z}^d continuos actions to measure preserving actions. The topologica Friedland entropy is ... 1answer 131 views ### L_p(I,Y)^\perp=L_q(I,Y^\perp)? Let X be a Banach space and Y be a closed subspace of X. For 1<p<\infty consider the p-th power Bochner Integrable functions which takes values in X and defined on the unit interval ... 1answer 255 views ### Transport of measure Let's disintegrate \mu and \nu, two probabilities on \mathbb{R}^{d} , according to$$ \pi_{k} (x_{1},...,x_{d}) = (x_{k},...,x_{d}) $$We get a family of measures and each measure \mu_{k,d}^{+... 2answers 326 views ### Supremum of continuous functions and essential supremum of continuous functions Suppose that (X,d) is a Polish metric space and A is a set of continuous bounded functions f:X\to \mathbb{R}. Suppose that \mu:X\to[0,1] is a Borel probability measure. Define$$\sup A:X\to ...
136 views

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 $X$ ...
131 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 $\Phi(t,x)$ ...
77 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 ...
535 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 ...
100 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 ...
73 views

### Implications of a Regularity Condition for Functions

$\newcommand{\essSup}{\mathop{\rm sup_{ess}}\nolimits}$ What can be concluded from the fact, that $f: X\ni x\mapsto f(x)\in [a,b]\subset\mathbb{R}\setminus\lbrace{-\infty,+\infty\rbrace}$ ...
181 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 ...
104 views

285 views

859 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 ...
240 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 $f\in F$ ...
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 ...