Questions tagged [measure-theory]

Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

10,799 questions with no upvoted or accepted answers
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Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
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Minimum area contained between measurable set and translate by $\lambda$: A strengthening of 2018 USA TSTST #9

Question Given $\lambda\in\mathbb{R}^+$, what is the smallest possible $c$ for which, given any measurable region $\mathcal{P}$ in the plane with measure $1$, there always exists a vector $\mathbf{v}$...
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Is it true that Lebesgue's differentiation theorem follows from Lebesgue's density theorem?

Let $(X,d)$ be a separable complete metric space and $\mu$ a probability measure on the Borel subsets of $(X,d)$. Suppose that the Lebesgue's density theorem holds, i.e. that for each Borel set $A$ of ...
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Forward characterization of measurable functions?

In topology, the standard definition of continuity works in the "backward" direction, since it puts a condition on the pre-images of a function rather than images: $f$ is continuous if the ...
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A characterization of the "Direct Integral" construction in terms of the properties it satisfies?

Fortunately there's a wonderful thing called the "direct integral" which enables one to make sense of direct sums of uncountably infinite families of Hilbert spaces. Unfortunately I've tried to read ...
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Cauchy-Formula for Repeated Lebesgue-Integration

Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given. Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
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Radon-Nikodym derivative of SDE law

Let $(\Omega, (\mathcal F, \mathbb P)$ be a probability space, $T$ some index set, and $(S, \Sigma)$ a measurable space. Let $X^f$ and $X^g : T × \Omega \to S$ be stochastic processes defined by the ...
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Stone Duality: What are $\sigma$-Algebras Dual To?

Stone duality, one of many dualities between certain lattices and certain topological spaces, asserts that there is a contravariant categorical equivalence between the category $\text{Bool}$ of ...
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Is Hilbert's space-filling curve measure preserving?

Say $f_n:[0,1]\to [0,1]^d$ is the $n$-th iteration of a $d$-dimensional Hilbert curve touring its range. Is it true that for any open $S\subset [0,1]^d$, then amount of time $f_n$ spends in $S$ is ...
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Intuition regarding the $\sigma$ algebra of the past (stopping times)

Let $(\mathcal F_n)$ be a filtration and $\tau$ be a stopping time with values in $\mathbb N \cup \{\infty\}$. Let $\mathcal F_\infty$ be the sigma-algebra generated by $\cup_n \mathcal F_n$. ...
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Proving $\lim\limits_{n\to \infty}\int\limits_{0}^{1} f_n(x)dx=0$

If $\{f_n\}$ is a sequence of continuous functions on $[0,1]$ such that $0\leq f_n\leq 1$ and such that $f_n(x)\to 0$ as $n\to \infty$, for every $x\in [0,1]$, then \lim\limits_{n\to \infty}\int\...
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This is a proof-verification request; I’m also recording this proof for my own later reference. Any feedback is appreciated. Claim: Let $X$ and $Y$ be independent, real-valued random variables on ...