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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2answers
44 views

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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1answer
38 views

$\lim_{p \to \infty} \|f\|_p = \|f\|_{\infty}$: 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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1answer
25 views

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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1answer
48 views

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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1answer
20 views

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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2answers
72 views

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 ...
2
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1answer
25 views

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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1answer
45 views

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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0answers
62 views

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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1answer
39 views

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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2answers
218 views

$f : \mathbb{R} \to \mathbb{R}$ (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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1answer
77 views

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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0answers
24 views

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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1answer
99 views

Measurable Functions with Common Sigma Sub-Algebras

Let $X:\mathbb{R}\rightarrow\mathbb{R}$ be a non-constant function, measurable with respect to the Borel-Algebra $\mathcal{B}$ and $\sigma(X)$ the sigma-algebra generated by $X$. Let $\mathcal{A}\...
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0answers
34 views

Prove that $f$ is measurable.

Let $u$ be a subharmonic function on the disk $\Delta$ centered at the origin with radius $\rho$. $u$ is not identically $-\infty$. Define $f:\Delta\times [0,2\pi]\rightarrow [-\infty, \...
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0answers
20 views

Does a linear operator on probability measures determine a Markov kernel?

Let $(\Omega, \mathcal{F})$ be a measurable space. Let $M$ be a linear operator on the space of probability measures on $(\Omega, \mathcal{F})$, i.e. for $\alpha \in [0,1]$ and probability measures $...
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1answer
65 views

Will this integral be progressively measurable?

Assume you have a function: $F(t,x,\omega)$: $[0,T]\times E\times \Omega \rightarrow \mathbb{R}$, which is predictable (predictable is explained below). Each of the three spaces can be viewed as 3 ...
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2answers
36 views

Arrow in Sigma Notation?

Working through a bioinformatics textbook currently, when I ran into this set of sigma equations: see the picture linked below. My interpretation of it is that it means if the sum from j to Si(t+1) of ...
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1answer
91 views

Differentiation under the integral sign for volumes in higher dimensions

Consider a smooth convex/compact domain $D\subset \mathbb{R}^n$ and a smooth, concave function $F:D\to \mathbb{R}$. Then we can define the function that simply takes the volume of the upper contour ...
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3answers
96 views

An equality concerning the Lebesgue integral

Let $f:X\mapsto[0,+\infty)$ be a non-negative measurable function defined on the space $X$, endowed with the complete $\sigma$-additive, $\sigma$-finite, measure $\mu$ defined on the $\sigma$-algebra ...
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1answer
42 views

Integral over all internal points

Let $X$ be some subset of real numbers, and $f$ a real-valued function on $X$. If $ f(x) \geq 0$ for every point $x\in {X}$, then also: $$\int_{x\in X} f(x)dx \geq 0.$$ What if $ f(x) \geq 0$ only ...
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1answer
30 views

How can I prove that an ultrafilter induces a finitely additive measure?

If $\mathcal{U}$ is an ultrafilter on a set $X$, it can be defined a function $\mu_{\mathcal{U}}\colon\mathcal{P}(X)\to\{0,1\}$ such that, for all $A\subseteq X$ it holds $\mu_{\mathcal{U}}(A)=1$ iff $...
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1answer
20 views

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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1answer
50 views

Real Analysis, Folland Theorem 2.41/Exercise 53 The $n$ dimensional Lebesgue integral

Theorem 2.41/Exercise 53 - If $f\in L^1(m)$ and $\epsilon > 0$, there is a simple function $\phi = \sum_{1}^{n}a_j\chi_{R_j}$ where each $R_j$ is a product of intervals, such that $\int |f - \phi| &...
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1answer
44 views

The $n$-dimensional Lebesgue Integral Properties

Background Information: In the theorem here we fix a complete Lebesgue-Stiltjes measure $\mu$ on $\mathbb{R}$ associated to the increasing right continuous function $F$, and we denote by $M_{\mu}$ ...
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1answer
50 views

Problem 5, Chapter 3 from Stein and Shakarchi's “Real Analysis” on a version of the FTC

The problem reads: Suppose that $F$ is continuous on $[a,b]$, $F'(x)$ exists for every $x\in(a,b)$, and $F'(x)$ is integrable. Then $F$ is absolutely continuous and $$F(b)-F(a)=\int_a^b F'(x)\,...
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2answers
79 views

Equivalent random variables and sigma algebras

Consider two random variables $X$ and $Y$ defined on the same probability space $(\Omega,\sigma,P)$. We know that they are equivalent in the sense that $P(\{X \ne Y\})=0$. Let $A_X$ and $A_Y$ be the ...
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0answers
30 views

Why $\lim_{x\to \infty }Mf(x)=0$ where $Mf$ is the Hardy-Littlewood maximal function.

In a course it is written that since $Mf(x)$ decay to zero at infinity, the measure or $\left\{x\in\mathbb R^n\mid\left|Mf(x)\right|>\lambda\right\}$ is finite. I was looking for such a result on ...
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1answer
60 views

Measurable non-negative function is infinite linear combination of $\chi$-functions

Let $(X,\mathfrak{M},\mu)$ is measure space. Suppose that $f\geqslant 0$ is measurable function on $X$. We know that exists the increasing sequence of measurable simple functions $\{s_n\}$ such that $...
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1answer
67 views

Drawing numbers on a plane *uniformly*

I am not being very precise here, because I do not know what would be the more precise terminology. I would definitely appreciate any comments on my method and my terminology. By choosing uniformly ...
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0answers
18 views

When is the product of integrable functions integrable over the product space?

In Stein & Shakarchi's Fourier Analysis, p. 46, a change in order of integration in $\int\int f(y)g(x-y)e^{-inx}dydx$ for $f$ and $g$ Riemann-integrable is justified by adding the ad-hoc ...
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0answers
38 views

Do $\limsup A_n$ and $\liminf A_n$ always have probability 1 or 0?

Is it right that, for a sequence of events $(A_n)_{n}$, $\limsup A_n$ and $\liminf A_n$ have probability $1$ or $0$?\ My idea for $\limsup$ is for example the following: $$P(\limsup A_n) = \mathbb{E}(...
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0answers
23 views

*-homomorphisms on $L^{\infty}(X,\mu)$ implemented by measurable maps

Let $(X,\mu)$ and $(Y,\nu)$ be two probability measure spaces. To any measurable map $T:(X,\mu)\to (Y,\nu)$ with $\mu=\nu\circ T$, the following injective $*$-homomorphism is induced: $$\rho_T:L^{\...
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1answer
40 views

Total variation defined as Integral

Let $\phi: [a,b]\to \mathbb{R}$ a function absolutely continuous with derivative continuous. Show that $TV(\phi, [a,b])=T_a^{b}(\phi)=\displaystyle\int_a^{b} \left|\phi'(x)\right|dx.$ Hello, my ...
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1answer
33 views

Is the supremum of an almost surely continuous stochastic process measurable?

Let's take a stochastic process $(X_t)_{0\leq t \leq 1}$ and assume that the sample paths are almost surely continuous. Let us define $S \equiv \sup_{t \in [0,1]} X_t$. How can we show that $S$ is ...
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0answers
31 views

Invariant measure of Poincare map

Let $M$ be a smooth manifold and let $v$ be a tangent vector field on $M$. Consider a system of ordinary differential equations $$ \dot x = v(x), $$ in local coordinates $x = (x_1, \ldots, x_n)$. ...
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2answers
73 views

Lusin's theorem from Rudin RCA

Hi! Let me ask questions on Lusin's theorem from Rudin's RCA. $1)$ As we know $s_n=\varphi_n \circ f$ (from Theorem 1.17) then $$2^nt_n(x)=2^n(s_n(x)-s_{n-1}(x))=2^n(\varphi_n \circ f(x)-\varphi_{n-1}...
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1answer
41 views

Separable metric space with no isolated points.

Let $(X,d)$ be a separable metric space with no isolated points and $(X,\mathcal{B}(X),\mu )$ is a measure space such that $\mu(X)<\infty$. How to prove that for all $\epsilon >0$, there exists ...
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4answers
822 views

Most functions are measurable

My professor once said that if you did not use the axiom of choice to build a function $f : \mathbb{R}^n \to \mathbb{R}^m$, then it is Lebesgue measurable. To what extent this is true?
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1answer
47 views

Use Poincare Recurrence to show existence of $n$

Suppose $A\subset \mathbb N$ such that $d(A)=\lim_{n\to\infty}\dfrac{|A\cap [1,n]|}{n}>0$. Then show there exists $n\in\mathbb N$ such that $\overline{d}(A\cap (A-n))>0$ where $\overline{d}(B)=\...
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1answer
31 views

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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1answer
54 views

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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1answer
28 views

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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2answers
49 views

Real Analysis, Folland Problem 2.4.33 Modes of Convergence

Background Information: Theorem 2.30 - Suppose that $\{f_n\}$ is Cauchy in measure. Then there is a measurable function $f$ such that $f_n\rightarrow f$ in measure, and there is a subsequence $\{...
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1answer
34 views

A graph of any function has content zero?

I know that if $f$ is integrable them his graph have content zero, but if we consider $f : \mathbb{R} \rightarrow \mathbb{R}$ as $f(x) = 1$ if $x \in \mathbb{Q}$ and $f(x) = 0$ if $x \in \mathbb{R} - \...
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1answer
44 views

Integrals over subset of measure space

Let $(X, \mathcal{M}, \mu)$ be a measure space. Suppose $E \in \mathcal{M}$ and $f \in L^+$ where $L^+$ is a space of measurable functions from $X$ to $[0, \infty]$. $\int_E f$ is defined by $\int_X f\...
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1answer
23 views

Strongly measurable implies Borel measurable in separable space

Let $(M,\mu)$ be a measure space, $X$ be a Banach space, $f: M \to X$ be a function. $f$ is said to be strongly measurable if there is a sequence of simple functions $\{f_n\}\to f$ pointwisely a.e.. $...
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2answers
30 views

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. ...
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0answers
69 views

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 $f_n$ ...