# Tagged Questions

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

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### Direct image of a sigma algebra

Let $(X,\cal{A}$) and $(X',\cal{A}')$ be two measurable spaces and $T:X\to X'$ a bijective measurable map. The claim is that $T(\cal{A})$ is a sigma algebra on $X' \iff T^{-1}:X'\to X$ is measurable....
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### The Jordan Decomposition Theorem, Folland

The Jordan Decomposition Theorem - If $\nu$ is a signed measure, there exists unique positive measures $\nu^+$ and $\nu^-$ such that $\nu = \nu^+ - \nu^-$ and $\nu^+\perp \nu^-$. Attempted proof - ...
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### $\lim_{p \to \infty} \|f\|_p = \|f\|_{\infty}$: is convergence monotone when $\mu(X) \leq 1$?

This question is related to Exercise 3.3.7(b) in Cohn, Measure Theory, 2nd edition, which reads as follows: Let $(X, \mathcal A, \mu)$ be a finite measure space, and let $f$ be an $\mathcal A$-...
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### Proving the Hahn Decompostion Theorem from Folland

The Hahn Decomposition Theorem - If $\nu$ is a signed measure on $(X,M)$, there exists a positive set $P$ and a negative set $N$ for $\nu$ such that $P\cup N = X$ and $P\cap N = \emptyset$. If $P',N'$ ...
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### For which $p$ is $\frac{1}{x^a+x^b}$ in $\cal{L}^p$?

Let $f(x)=\frac{1}{x^a+x^b}$ with $x,a,b>0$. For which $p\ge1$ is $f$ in $\cal{L}^p(\lambda)$ over the interval $(0,\infty)$? Here $\lambda$ is the one dimensional Lebesgue measure. Attempt: We ...
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### Showing left continuity of this distribution function

Suppose $(X,\cal{A},\mu)$ is a $\sigma$-finite measure space and $u$ is a nonnegative measurable function. Then the function $F:\mathbb{R} \to \mathbb{\bar{R}}$ given by $t\mapsto \mu(\{x: u\ge t\})$ ...
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### What is the diagonal principle?

I'm guessing I'm supposed to show that there exists a subsequence that converges in probability quickly (hypothesis of $(c)$). What is the diagonal principle? Is that related to Cantor's ...
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### Continuity of a characteristic function of a translated set

Let $E \subseteq \mathbb{R}$ be a measurable set. Is it true that $\chi_{E+t}(x) \rightarrow \chi_{E}(x)$ as $t \rightarrow 0$, where $E+t = \{x+t \, | \, x \in E\}$ for each $t \in \mathbb{R}$, ...
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### Lusin's Theorem, Modes of Convergence

Background Information: Theorem 1.18 - If $E\in M_{\mu}$, then \begin{align*} \mu(E) &= \inf\{\mu(U):E\subset U, U \ \text{open}\}\\ &=\sup\{\mu(K):E\subset K, K \ \text{compact}\}\end{...
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### Show that $\int_{E}{F(x,t)}d\mu\otimes\lambda=\int_{X}{\int_{[\varphi_1(x),\varphi_2(x)]}{F(x,t)d\lambda(t)d\mu(x)}}$

Let $(X,\mathcal{F},\mu)$ be a $\sigma-$finite measure space. Let $\varphi_1,\varphi_2:X\to\mathbb{R}$ functions in $\mathcal{M}(X,\mathcal{F},\mathbb{R})$ such that $\varphi_1(x)\leq\varphi_2(x)$ for ...
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### Halving a measureable set

Let $X$ be a set with a finite Lebesgue measure (e.g. a subset of the unit interval), and $u,v$ two measures such that: $$v(X) > u(X)$$ Does there exist a subset $Y\subset X$, with Lebesgue ...
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### $f : \mathbb{R} \to \mathbb{R}$ (Lipschitz) continuous implies $f(A)$ is Borel for all Borel $A$.

Full question: Let $(\mathbb{R}, \mathfrak{M}, m)$ denote the measure space $\mathbb{R}$ equipped with the Borel $\sigma$-algebra and the Lebesgue measure. Suppose $f : \mathbb{R} \to \mathbb{R}$ is ...
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### Construction of a measure space from some weird functional

Here is the complete problem I am trying to solve, but currently, I am just interested in proving that $\Sigma$ is a $\sigma$-algebra. Let $X$ be a set and $\phi: 2^X \to [0, \infty]$ be a ...
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### Lower semicontinuous function is measurable

Let $(X, \mathfrak{M},\mu)$ be measure space such that $\mathfrak{M}$ contains all Borel sets of $X$. Let $f:X\to \mathbb{R}$ be lower semicontinuous function (LSC). Prove that $f$ is measurable. ...
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### Cancelling measures on both sides

Let $(X,\mathscr{A},\mu)$ be a measure space. Suppose I have $\mu(A)-\mu(B)\ge\mu(A)$ with $\mu(A)=+\infty$. Am I still allowed to cancel the $\mu(A)$'s from both sides and conclude that $\mu(B)=0$?
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### Theorem 2.22 from RCA Rudin

I read this interesting result from Rudin's book but I would like to clarify some confusing moments. As I understood $(\mathbb{R}^1, +)$ is group and $(\mathbb{Q}, +)$ is subgroup. He considers ...
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### Real Analysis, Folland Problem 2.4.42, counting measure with convergence in measure

Problem 2.4.42 - Let $\mu$ be counting measure on $\mathbb{N}$. Then $f_n\rightarrow f$ in measure if and only if $f_n\rightarrow f$ uniformly. Attempted proof - Suppose that $\mu$ is a counting ...
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### Can anyone explain an example related to vague convergence?

Let {$X_n$} be a sequence of random variable and {$\mu_n$} be a sequence of measures induced by {$X_n$} such that $\mu_n(B) = \mathbf{P}(X_n^{-1}(B))$. Suppose $X_n$ following a uniform distribution ...
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### Does the collection $\{ \ [n, \infty) \ \mid \ n \in \Bbb N\}$ generate the Borel $\sigma$-algebra on $\Bbb R$?
Does the collection $\mathfrak C = \{ \ [n, \infty) \ \mid \ n \in \Bbb N\}$ generate the Borel $\sigma$-algebra on $\Bbb R$? I came across this problem in a paper. It seems like the answer is No. ...
### How to prove $f_n \in L^1$
I was trying to build a scheme to solve this kind of question: Let $D$ be a domain of $\Bbb R^n$ and $f_n\colon D \to \Bbb R$. Say if $f_n \in L^1(D)$. First of all I need to check that both $f_n$ ...