# Questions tagged [measurable-functions]

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### 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 $\mathcal M$ ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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-...
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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 $$t \mapsto f(t,x) \text{ is measurable}$$ $$x \... 1answer 105 views ### Measurable selection for argmin to distance Let Y be a Banach space and equip Y with the weak topology. Now, let X be a closed, bounded, and convex subset of Y and suppose that the relative (weak) topology on X is metrizable with ... 2answers 176 views ### Sharp assumption for preserving Lebesgue measurability by left composition Let g: [0, 1] \to \mathbb R be a Lebesgue-measurable function (in the classical sense: the inverse images of Borel sets are Lebesgue-measurable). It is a classical fact in analysis that f \circ g ... 0answers 78 views ### Measurability of infimum function In Theorem 14.37 of Variational Analysis by Rockafellar and Wets, it shows that for any normal integrand f:T\times \mathbb{R}\to\mathbb{R}, the function p:T\to \mathbb{R} given by p(t):=\inf f(t,⋅... 1answer 119 views ### Friedland metric entropy I was asking if it is possible to extend the definition of topological Friedland entropy for \mathbb{Z}^d continuos actions to measure preserving actions. The topologica Friedland entropy is ... 1answer 142 views ### L_p(I,Y)^\perp=L_q(I,Y^\perp)? Let X be a Banach space and Y be a closed subspace of X. For 1<p<\infty consider the p-th power Bochner Integrable functions which takes values in X and defined on the unit interval ... 1answer 347 views ### Transport of measure Let's disintegrate \mu and \nu, two probabilities on \mathbb{R}^{d} , according to$$ \pi_{k} (x_{1},...,x_{d}) = (x_{k},...,x_{d}) $$We get a family of measures and each measure \mu_{k,d}^{+... 2answers 821 views ### Supremum of continuous functions and essential supremum of continuous functions Suppose that (X,d) is a Polish metric space and A is a set of continuous bounded functions f:X\to \mathbb{R}. Suppose that \mu:X\to[0,1] is a Borel probability measure. Define$$\sup A:X\to ...
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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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### 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 $\Phi(t,x)$ ...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...