# Questions tagged [measure-theory]

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

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### Density and Outer Measure

If $A \subset \mathbb{R}^{n}$ is Lebesgue measurable, then we see that $x \in A$ is a point of density if $$\lim_{r \to 0} \frac{\mu(A \cap B(x,r))}{\mu(B(x,r))} =1$$ The Lebesgue Density Theorem ...
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### Sigma algebra generated by product of random variable

This question relates to this. Let $X,Y$ be two (real) random variables on $(\Omega,F)$. It was stated that $$\sigma(XY)\subseteq \sigma(X,Y).$$ The argument used is as follows: let $f(x,y)=xy$ be a ...
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### $\int_{E}^{*}{h(t,f_n(t)d\mu(t)}=\int_{E}{a_0\big(t,f_n(t)\big)d\mu(t)}$

Let $(T, \mathcal{A}, \mu)$ be an arbitrary measure space, and $E$ a convex subset of a Hausdorff topological vector space. Any function from $T\times E$ into $( - \infty, + \infty]$ is be called ...
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### How to prove this function is L-continuous almost-everywhere?

Definition: Let $(X, \mathcal{M}, \mu)$ a measure space. Some property P (in this case, the continuity) is said to be satisfied almost everywhere in X if there ...
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### $\sigma$-algebra generated by a finite family of sets

The question is the exercise 1.7 of "Measure theory and probability theory" by Krishna and Soumendra. Let $\mathcal{B}=\{B_1, ..., B_k\}$ with $B_i \subseteq \Omega$ and $\bigcup_{i=1}^kB_i=\Omega$. ...
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### Reference to Lebesgue space on manifolds [closed]

I am looking for a reference to the Lebesgue space $L^p(M)$, where $M$ is a manifold.
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### Why does linear combination of dirichlet function is not a step function？

I know that Dirichlet function is not Riemann integrable but only Lebesgue integrable which has measure zero. I am having issue to understand the example that a simple function which is not step ...
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### Can we say that $\frac{1}{n}\sum_{i=1}^{n}{f_i(t)\to 0 }~~\text{a.e}$?

Let $(E,\mathcal {A},\mu)$ be a finite measure space and $\{f_n\}$ be a sequence bounded in $L^1$, such that: $$f_n(t)\to 0 ~~\text{a.e and in } L^1$$ Can we say that  \frac{1}{n}\sum_{i=1}^{n}{...
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### Sigma-algebra generated by a sequence with zeros

Let $\{x_i\}_{i\in\mathbb{N}}$ be a sequence of zero mean random variables on a probability space $(\Omega, F, P)$. Suppose that there is some $i_0>0$ such that $x_i=0$ for all $i\geq i_0$. Does ...
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### Doubt about the computation of the volume of a sphere

Consider the measure $d\Omega_{n+1}$ as a invariant measure induced by the Lebesgue measure on the unit sphere $\mathbb{S}\subset \mathbb{R}^{n+1}$. I want to calculate $d\Omega_{n+1}(\mathbb{S})$. I ...
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### Is Banach–Tarski paradox false without axiom of choice? [duplicate]

I know that you need axiom of choice to prove Banach–Tarski paradox. But what happens with paradox when we remove axiom of choice? Does theorem become false? Or is there just no proof of it without ...
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### Reference for the Dunford-Pettis Theorem.

Does anyone have a book reference for the Dunford-Pettis theorem as stated in the Wikipedia article https://en.wikipedia.org/wiki/Uniform_integrability.
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### Hilbert space version of the notion of conditionally weakly-mixing functions.

$\newcommand{\norm}{\|#1\|}$ $\newcommand{\ab}{\langle #1\rangle}$ $\newcommand{\mr}{\mathscr}$ $\newcommand{\mc}{\mathcal}$ $\newcommand{\E}{\mathbb E}$ $\newcommand{\C}{\mathbf C}$ ...
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### Lp Space: an intuitive understanding.

Would someone be willing to give me an intuitive understanding of the Lp space? I have several analysis books which (seemingly) approach the topic differently, which confuses me more, and a simple ...
I am reading Measure Theory by Halmos, and I was wondering if someone could help me on the following definition: If there exists a mean fundamental sequence $\left\{f_{n}\right\}$ of integrable ...
I'm self-studying measure theory. I stumbled upon this statement: If $X \subset \mathbb{R}$ and $f: X \to \mathbb{R}$ is a function, then $f$ is a Borel measurable function if and only if \$f^{-1}((...