# Questions tagged [measure-theory]

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

23,131 questions
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### Show that finitely many $A_n$ will occur

I tried this problem quite a bit but went nowhere. I wish to solve this using set-theoretic algebra. Problem statement Let $A_n$, $n \geq 1$ be a sequence of events such that $\mathbb P(A_n) \to 0$ ...
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### Is this “inverse” statement of Fubini’s theorem true?

Consider $f:\mathbb{R}^{d_1}\times\mathbb{R}^{d_2}\to\mathbb{R}$ , Fubini’s theorem says if $f$ is Lebesgue integrable than we can integrate first over $x$ and then over $y$ and get the same value as ...
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### Are $L_p$ norm and discrete $L_p$ norm comparable?

Are there any estimates on how a $L_p$ norm (say for a compact set in $\mathbb{R}$) is related to a discrete $L_p$ norm, where we could for example consider the Jackson integral on this compact set. ...
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### Absolutely continuous measure and equality to $0$

In the context of proving that the data processing inequality for $f$-divergences hold for any Markov kernel I am interested in the following statement If $\mu$ and $\nu$ are two probability ...
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### Example of countably additive function from a Boolean algebra to [0,infinity] that is not finitely additive

This example is implied to exist by Tao’s undergrad measure theory text, exercise 1.7.4, with the claim that further assuming that the empty set getting mapped to 0 prevents such a function from ...
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### Show that the set $A = \bigcup\limits_{k \in \mathbb{Z}}[3^k - 2^{-|k|}, 3^k)$ is Lebesgue measurable and find its measure.

Show that the set $A = \bigcup\limits_{k \in \mathbb{Z}}[3^k - 2^{-|k|}, 3^k)$ is Lebesgue measurable and find its measure. I'm having trouble with both parts of the question. First Part: By ...
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### Weak Convergence of Probability Measures (Proving integrals converge if measures do)

Note: I am doing this question just for fun, not for hw. Question: Fix any dense subset $G$ of the unit ball of $C^{0}(M)$. Here $C^0(M)$ refers to the space of continuous functions defined on $M$, a ...
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### Integration w.r.t. pushforward measure

Problem: Let $\phi \colon [0,1] \to [0,1]$ be a continuous function and let $\mu$ be a Borel probability measure on $[0,1]$. Suppose $\mu(\phi^{-1}(E)) = 0$ for every Borel set $E \subseteq [0,1]$ ...
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### Conditional Distributions and Rationals

If $U$ is uniformly distributed on $[0, 1]$, what is the conditional distribution of $U$ given that $U$ is rational? Intuitively, it would be uniform, but we cannot have a uniform distribution over a ...
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### How to see the convergence/divergence of this integral [duplicate]

I have a question, I want to know for which values of $p\in \mathbb{R}$ this integral have finite value: $$\int_{0}^{\infty} \frac{\sin{x}}{x^p}dx$$ I have shown that for $p=1$ the integral is finite ...
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### how to show that $A_{k-1} \subset 3A_{k}$ where $A_k$ are iterations of the Cantor set ???

For example $A_1 = [0,1/3] \cup [2/3 , 1]$ $A_2 = [0, 1/9] \cup[2/9 ,1/3] \cup [2/3 ,7/9] \cup [ 8/9 ,1]$ Note that $3 (A_2) = A_1 \cup D$ , for some set $D$ In the general case, I want to ...
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### Royden Chapter 6 [on hold]

I'm really stuck on this question. Any help would be appreciated.
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### An application of Kolmogorov $0$-$1$ law

I need to approve or to disapprove the following statement : if $(X_n)_{n \in \mathbb{N}}$ is a sequence of independent random variables and identically distributed, and $(u_n)_{n \in \mathbb{N}}$ is ...
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### For every $\epsilon\gt 0$, $|a-b|<\epsilon$ ,then b=a .

I have done a proof by myself but not sure about it proof: $|b-a|<\epsilon$ =$a-\epsilon$
### Lebesgue Measure in $\Bbb{R}^k$ is invariant under isometries
I'm studying Lebesgue Measure. I have a problem on proving that Lebesgue Measure in $\Bbb{R}^k$ is invariant under isometries Here is my work so far. Let $T$ $\mathbb{R}^k \to \mathbb{R}^k$ is an ...