Questions tagged [measurable-functions]

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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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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-...
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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 $f: X\ni x\mapsto f(x)\in [a,b]\subset\mathbb{R}\setminus\lbrace{-\infty,+\infty\rbrace}$ ...
121 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 ...
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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 ...
238 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 ...