# Tagged Questions

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

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### How to construct the appropriate measurable set.

I searched but couldn't find an answer similar to this topic, apologies if I missed it. Here goes; Let $A_j \subset [0,1], j=1,2,...$ such that each $A_j$ has Lebesgue measure $\mu(A_j)\geq 1/2$. ...
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### In search for a sequence of r.v. with particular conditions

I am looking for a sequence of real random variable $(X_n)_{n\geq 0}$ on a probability space $(\Omega,\mathcal F, \mathbb P)$ such that : $\forall n >0,$ $X_{n+1}-X_{n} <+\infty$ a.s. It ...
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### Measure of a finite union of almost disjoint intervals

This is my first time seen real analysis and measure theory, so I'm trying to study in advanced. In zygmund's book, there is a lemma: if $I_k$ is a finite collection of non overlapping intervals, then ...
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### Following conditions for convergence of measures equivalent

Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0, 1]$. Let $\mu_n$ be a sequence of finite measures on $([0, 1], \mathcal{B})$ and let $\mu$ be another finite measure on $([0, 1], \mathcal{B})$. ...
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### Show that if $E$ is Jordan measurable then $m(A-B) \leq \epsilon$

I want to show that if $E$ is Jordan measurable then $m(A-B) \leq \epsilon$ where $A \subset E \subset B$. I think I have the right ideas but feel I am missing some details. I'd like some feedback ...
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### First uncountable ordinal

I am a beginner of ordinals and I don't know any powerful techniques in it. I come across with a problem about the first uncountable ordinal like this. Let $X$ be a set of uncountable cardinality. ...
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Suppose that $\{r_j\}_j$ is a sequence of positive real numbers, and $\{x_j\}_j$ is a sequence in $\mathbb{R}^n$. Suppose also that there are $r \geq 0$ and $x \in \mathbb{R}^n$, such that $$\lim_{j\... 0answers 45 views ### Discrete random variable whose cdf is not a step function [on hold] Let, (\Omega,\mathcal{F},P) be a probability space and X:\Omega \rightarrow \mathbb{R} be a random variable. Let F_{X} (x) be the cumulative distribution function of X. Show that if F_{X} (x)... 1answer 387 views ### Sub sigma algebra and probability spaces — definition I am reading this book and I am a bit lost with the definitions because they are not provided and I can't seem to find it online: Let L_2(\Omega,A,P) be a probability space such that f \in L_2 ... 0answers 13 views ### Why in the definition of multiple integrals on subset A\subset \mathbb{R}^n it is required that A is measurable? I'm new with the study of multiple integrals. I think I understood the topics of Peano–Jordan measure. A multiple integral is defined on a measurable (and limited) subset A\subset \mathbb{R}^n, ... 1answer 29 views ### Integral of Simple Functions converges to Integral of Measurable Function Let f be a measurable function and E_{n,m} = \{x : \frac{m}{2^n} \leq f(x) < \frac{m+1}{2^n} \}. Prove:$$\lim_{n \to \infty} \sum_{m=1}^{\infty} \frac{m}{2^n} \mu(E_{n,m}) \to \int f \, d\mu$... 0answers 20 views ### What's$\{g(\theta^n x)\} $sequence called? Let$(S, A, µ)$be a probability space and$g$be a measurable function on it. Let$\theta$be a µ-measure preserving transformation on it. If$\theta$is a ergodic, what's$\{g(\theta^n x)\} $... 0answers 33 views ### Hölder inequality application to show that f=1 I want to proof that if$f \in L^{1}_{\mu}(\mathbb{R}), f > 0$continuous, satisfies$(\int_\mathbb{R} f(x)d\mu)^{3} \le \int_\mathbb{R} f(x)^{3sin^{2}(x)}d\mu * (\int_\mathbb{R}f(x)^{\frac 32cos^{...
If $A_i$ is a sequence of sets, define$$\liminf_i A_i = \bigcup_{j = 1}^\infty \bigcap_{i = j}^\infty A_i, \quad \limsup_i A_i = \bigcap_{j = 1}^\infty \bigcup_{i = j} A_i.$$Given a set $D$ define the ...