Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
3,228 questions
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Does anyone use measures that take values in real numbers and cardinal numbers?
The counting measure on $\mathbb{R}^n$ is a map that takes a subset $A$ of $\mathbb{R}^n$ and returns its cardinality if it is finite or the symbol $\infty$ if it is infinite.
So, if $A\subseteq\...
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Is there a increasing, convex, superlinear $f$ with $c_1 f(x)y \leq f(xy)\leq c_2 f(x)f(y)$ such that $\mathbb{E}[f(X)] < \infty$?
The following version of a de la Vallée Poussin - criterion would be very helpful to me if it would be true. Can you say something about the truth value or give a reference?
Given a positive random ...
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Function of two sets intersection
Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
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Minimal dominating measure for dominated Markov kernel
Let $(X,\mathcal X)$ and $(Y,\mathcal Y)$ be measurable spaces, $\pi \colon \mathcal Y\times X \to [0,1]$ a Markov kernel. We assume that it is measurably dominated, i.e. there is a $\sigma$-finite ...
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Measurability of $t \mapsto \int_A f(t, \omega)\mathbb{Q}_t(\mathrm{d}\omega)$ when $(t, \omega) \mapsto f(t, \omega)$ is not measurable in $t$
I have a Markov kernel $(t,A) \mapsto \mathbb{Q}_t(A)$ from a standard Borel space $(T, \mathcal{T})$ into another standard Borel space $(\Omega, \mathcal{F})$. Also, for $t \neq s$, $\mathbb{Q}_t \...
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Nonseparable Hoffmann-Jørgensen metric space
A metric space $(X,d)$ satisfies the Hoffmann-Jørgensen (HJ) property if for any two Borel measures $\mu_1,\mu_2$ we have that $\mu_1(B_r(x))=\mu_2(B_r(x))$ for all $r>0$ and $x\in X$ implies $\...
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Definable collections of non measurable sets of reals
Is there a definable (in Zermelo Fraenkel set theory with choice) collection of non measurable sets of reals of size continuum? More verbosely: Is there a class A = {x: \phi(x)} such that ZFC proves "...
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Disintegration of a push forward measure
Let $\mu$ and $\nu$ be two probability measures on $\mathbb{R}^{d}$.
Let $T : \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ be a measurable map such that $T_{\ast} \mu = \nu$.
I can disintegrate $\gamma :...
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Integral representation of Markov operators
On a measurable space $(E,\mathcal E)$, a stochastic kernel is a function $p\colon E\times \mathcal E\to [0,1]$ such that:
for each $x\in E$, the function $A\mapsto p(x,A)$ is a probability measure;
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Which sets are "persistently measurable"
Let $\mathcal{E}$ be the class of Lebesgue-measurable subsets of $\mathbb{R}$. The notion of $(\mathcal{E},\mathcal{E})$-measurable function is a bit pathological since lots of continuous functions ...
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Inverting the conditional expectation for some coupling
Let $\nu$ be a probability measure equivalent to $\mathbf{1}_{\mathbb{R}_+}(y) \, \lambda(dy)$. Let $\pi$ be a probability measure on $\mathbb{R}^2$ of second marginal $\nu$, such that $\nu(dy)$-a.e.,
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Let $k \mapsto f_k$ have nonnegative derivative in $L^0(\mu)$, then it is increasing almost everywhere
Let $\mu$ be a finite measure on some measurable space $(X, \Sigma)$ and consider the topological vector space $L^0(\mu)$ of all real-valued measurable functions on $X$ with respect to convergence in ...
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A question on the Banach space property of a rearrangement invariant function space
Consider a measure space $(S,\mu)$ and assume that $\mu(S)=1$. We consider the quantile function (or nonincreasing rearrangement) of a real valued function $f:S\to\mathbb{R}$ as the function
\begin{...
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Borel-Cantelli lemma for general measure spaces (those with infinite measure)
The Borel-Cantelli lemma is often stated for a probability space or spaces with finite measure.
But it seems to me that it still holds if the space $X$ is of infinite measure. I seem to be able to ...
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From weak to norm continuity: uniqueness of representing measures on $C_b(H)$
I am studying the problem of representing positive linear functionals on the space of bounded norm-continuous functions on a separable infinite-dimensional Hilbert space $H$. A known challenge is that ...
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