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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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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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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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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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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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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Defining Lebesgue non-measurable sets with countable information
Is there a formula $\phi$ in the language of set theory such that
$$
\text{ZFC proves } \exists x \in \mathbb{R}:\text{ the set }A_x:=\{y\in\mathbb{R}:\phi(x,y)\} \text{ is not Lebesgue measurable?}
$...
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Successive Riemann integrability of products of successively Riemann integrable functions
In teaching multivariable Riemann integration, I was trying to develop the theory of successive Riemann integrals (so all start with the one-dimensional case familiar to the students) as far as ...
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How to prove the convergence of the maximum point random variable of random concave function sequence？
I am really wondering how to prove this lemma from the book 'Counting processes and survival analysis'. No need for the first and second point, just the third point, why does the maximum random ...
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Consistency of a measure witnessing a strengthening of Freiling’s axiom of symmetry
I'm interested in Freiling's axiom of symmetry and I specifically wonder if it may be proven from more basic axioms about measures on $\mathbb R^n$, in the sense that there is a sequence of measures $\...
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Understanding the theorem from Kechris' book that Borel sets are mapped to Borel sets under injective functions
Theorem 15.1 in Classical Descriptive Set Theory by Kechris states:
(i) Let $X, Y$ be Polish spaces and $f:X\rightarrow Y$ be continuous. If $A\subseteq X$ is Borel and $f|_A$ is injective, then $f(A)$...
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When is a mapping that is both a measure isomorphism mod 0 and an order isomorphism unique mod 0?
Suppose we have two measure spaces, $(\Omega, \mathscr{F}, \mu)$ and $(\Psi, \mathscr{G}, \nu)$, with $\mu(\Omega) = \nu(\Psi) < \infty$, and we consider the set of measure isomorphisms mod 0 ...
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