# Questions tagged [measure-theory]

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

8,064 questions with no upvoted or accepted answers
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### Is this really a categorical approach to integration?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A ...
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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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### Erratum for Billingsley’s $\textit{Probability and Measure}$, Problem 32.13

This is a verification request for a counterexample that I think I have found for Problem 32.13 on page 427 in Patrick Billingsley’s Probability and Measure textbook (third edition, but the problem ...
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### Measure-driven differential equations

Background: I need some help to understand the concept behind measure-driven differential equations. The solution of an ordinary differential equation is continuous. In order to describe discontinuous ...
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### If the difference of two independent random variables has a mean, so does each variable

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 ...
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### On the Lebesgue measure of a cartesian product

If $X \subseteq \mathbb{R}^{l}$ and $Y \subseteq \mathbb{R}^{r}$ with $l + r = n$, is it true that $\lambda_{n}(X \times Y) = \lambda_{l}(X) \cdot \lambda_{r}(Y)$ (where $\lambda_{m}$ is the Lebesgue ...
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### Lebesgue Premeasure via Transfinite Induction

If $I=[a,b)$ we write $|I|=b-a$ for the length of $I$. Given a theorem of Caratheodory, the tricky part in showing the existence of Lebesgue measure is this: Lemma If $[0,1)$ is the disjoint union of ...
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### Can an arbitrary probability space be simulated by coin tosses?

Let $B=\left(\{\mathrm{heads},\mathrm{tails}\},\mathcal{P}\{\mathrm{heads},\mathrm{tails}\},\mu\right)$ be the probability space for a single fair coin toss. For any cardinal $\aleph$ let $B^\aleph$ ...
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### Clarification on product space, conditional probability measures

Suppose we have two random variables $X_i$ on the probability space $(\Omega_i, \sigma(\Omega_i),P_i)\; i = 1,2$. Now, $P_i$ is just a technical tool and we consider directly the distribution of $X_i$ ...
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### Monotonicity of measures

Let $\mu$ be a measure defined on $\Omega$. Then $\mu(A)\le \mu(B)$ for all $A\subset B\subset \Omega$. pf. Let $A\subset B$, let $C=A^c\cap B$. Then $A\cap C=\emptyset$ and $A\cup C = B$. By ...
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### Show that the union over a collection of compact cubes in $\mathbb{R}^n$ is Lebesgue measurable

Let $\mathcal{K}$ be a (not necessarily countable) collection of compact cubes in $\mathbb{R}^n$. Show that $\cup\{K:K\in \mathcal{K}\}$ is a Lebesgue set (Measurable with respect to the Lebesgue ...
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### What are some characterizations of the strong and total variation topologies on measures?

The Wikipedia article on convergence of measures defines three kinds of convergence: total variation, strong, and weak. For weak convergence, a number of equivalent formulations are given, but not for ...
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### Outer Measure of the complement of a Vitali Set in [0,1] equal to 1

I am trying to prove the first part of exercise 33, ch. 1 in Stein and Shakarchi (Real Analysis). I am running into some difficulties following the hint though. Here is the problem (note, $N$ is a ...
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### Lebesgue point of density on $[0,1]$ and Dynkin's theorem

The problem defines a density point $x\in[0,1]$ for a Borel set $A\subset [0,1]$ if $$\lim_{\varepsilon \rightarrow 0^+} \frac{\mu([x-\varepsilon,x+\varepsilon]\cap A)}{2\varepsilon}=1.$$Denote all ...
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### Infinite Burnside lemma

Burnside's lemma says that if a finite group $G$ acts on a finite set $X$ then the number of orbits equals the average number of fixed points: $$|X/G| = \frac{1}{|G|} \sum_{g \in G} |X^g|.$$ I was ...