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

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3 views

Convex conjugate of average Fisher information measure

What is a possible convex conjugate of the function $\rho \mapsto \int (\nabla \log \rho(x))^2 \rho(x) dx$? (Suppose $\rho$ is a sufficiently integrable probability density function on a $d$-...
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0answers
15 views

Show that if $E$ is Jordan measurable then $m(A-B) \leq \epsilon$

My idea: Let $\epsilon>0$ be given. Suppose $E$ is Jordan measurable. Then there exists $A$ and $B$ elementary with $A \subset E \subset B$ such that $$ \sup m(A) = \inf m(B) $$ and thus $$ m(A)...
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1answer
43 views

Is Fuzzy Set/Measure Theory an Active Area for Research?

I came across the notion of a fuzzy set the other day and since then, I've been reading about fuzzy measures and the Sugeno/Choquet integrals. While I certainly do not claim to have fully wrapped my ...
2
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2answers
33 views

Conditional expectation of a product XY given Z with Y independent of Z

Let $X,Y$ and $Z$ be integrable random variables s.t. $XY$ is integrable and $Y$ is independent of $Z$ . I was wondering if there are any helpful/common ways of rewriting $\mathbb{E}[XY\mid ...
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0answers
18 views

A result about weighted-sum of uniform random variables

Let $a_1,\ldots,a_m \in \mathbb{Z}$ and $U_1,\ldots,U_m$ be independent uniform random variables taking valules in $[0,1]^d$. Let $\mathcal{Z}$ be the support of the random variable $\sum_{i=1}^m a_i ...
1
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1answer
24 views

sum of two sequences of functions converging in measure still converges in measure

Suppose $f_n\to f$ in measure and $g_n\to g$ in measure. Can I claim that $(f_n+g_n)\to f+g$ in measure? Attempt at the proof: Since we know that $f_n$ and $g_n$ converge in measure respectively, we ...
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0answers
13 views

What is the condition for this limit to hold? (see question)

Let, for $n, m = 1,2,..., a_n(m)$ and $a_n$ be real numbers such that $a_n(m) \rightarrow a_n$ as $m \rightarrow \infty$. Use the dominated convergence theorem to formulate a condition under which $\...
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1answer
45 views

Lusin's Theorem: Can we assume nested sets?

This is the statement of Lusin's Theorem (taken from Royden): Let $f$ be a real-valued measurable function on $E$. Then for each $\epsilon>0$, there is a continuous function $g$ on $\mathbb{R}$ ...
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1answer
28 views

showing $\int f_n^+\to \int f^+$

Supose $(f_n)$ be such that $\int|f_n-f|\to 0$, where $(f_n)$ is Lebesgue integrable. Show that $\int_E f_n \to \int_E f$ for all Lebesgue measurable sets $E$, and furthermore that $\int f_n^+\to \int ...
4
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0answers
30 views

If $m^*([-n,n] \cap E) + m^*([-n,n] \setminus E) = 2n$ for all $n$, then $E$ is Lebesgue measurable

Let $E \subset \Bbb R$ and let $m^*$ denote the Lebesgue outer measure on $\Bbb R$. Show that if for all $n \in \Bbb N$, $m^*([-n,n] \cap E) + m^*([-n,n] \setminus E) = 2n$, then $E$ is Lebesgue ...
2
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1answer
23 views

How do I prove two definitions of the variation of a measure are equivalent?

Let $(X,\Sigma)$ be a measurable space and $\mu:\Sigma\rightarrow \mathbb{C}$ be a complex measure. Define $|\mu|(E)$ as the supremum of $\sum_{n=0}^\infty |\mu(E_n)|$ where $\{E_n\}$ is a mutually ...
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1answer
23 views

Hausdorff measure vs Lebesgue measure for a hypersurface in $\mathbb{R}^n$

Let $H$ be a compact smooth hypersurface with boundary in $\mathbb{R}^n$. We can compute the Lebesgue measure $\mathcal{L}(H)$ with respect to the induced Lebesgue measure coming from $\mathbb{R}^n$, ...
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0answers
36 views

Discrete random variable whose cdf is not a step function

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 $F_{X} (x)$...
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0answers
12 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 $A\subset \mathbb{R}^n$, ...
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0answers
17 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)\} $ ...
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2answers
51 views

another version of criterion for measurable set

Let $\mu^{\ast}$ be the outer measure on $R$.A collection $\left\{A_i\right\}$ is a partition of $R$ if $A_i \cap A_j=\phi$ if $i\neq j$ and $\bigcup^\infty_{i=1} A_i=R$. Prove that all sets on the ...
2
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1answer
29 views

An application of Egoroff' theorem

Let $\left\{f_{n}\right\}$ be a sequence of measurable functions on the real line $\mathbb{R}$, and $f_n\rightarrow f$ almost everywhere. Prove that there exists a sequence of measurable sets $\left\{...
2
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1answer
38 views

Sequence of integrable function with $\sum_{n=1}^\infty \|f_n\|_1<\infty$. Show that $\sum_{n=1}^\infty f_n$ converges a.e. and is integrable.

Let $\{f_{n}\}$ be a sequence of functions in $L^1(\mathbb{R})$ such that $\displaystyle \sum_{n=1}^\infty\|f\|_{1}<\infty.$ Show that $$f(x): = \sum_{n=1}^\infty f_n(x)\text{ converges a.e., }\, f\...
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2answers
54 views

Characteristic function of a ball with radius $r$ centered at $x$

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\...
2
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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$...
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1answer
44 views

Fubini's theorem for a measure on product space, which is not a product of measures

Let $X,Y$ be some nice measurable spaces (I'm interested in $[0,1]$ so we can assume compact, etc.). Let $\mu$ be a measure on $X\times Y$ (again, assume it's a nice probability measure, or even ...
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0answers
46 views

Is a measure on product space necessarily a product of measures?

Let $X,Y$ be some nice measureable spaces (i'm interested in $[0,1]$ so we can assume compact, etc.). let $\mu$ be a measure on $X\times Y$.(again, assume it's nice, i.e. probability measure. anything ...
2
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0answers
20 views

Proving that $A \mapsto \sup\{ \mu E \mid A \supset E \in \Sigma, \mu E < \infty\}$ is an inner measure

Let $(X,\Sigma, \mu)$ be a measure space and define $m: 2^X \to [0,\infty]$ by $m A = \sup\{ \mu E \mid A \supset E \in \Sigma, \mu E < \infty\}$. Show that $m$ is an inner measure. There are $4$ ...
2
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1answer
12 views

The uniform measure on $A^{\mathbb{Z}}$

Let A be a finite set and consider $A^{\mathbb{Z}}$. A is equipped with the discrete topology and $A^{\mathbb{Z}}$ with the associated product topology. By $\mu_u$, denote the uniform measure on $A^{\...
1
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1answer
28 views

Lebesgue decomposition and Radon-Nikodym derivative given a function.

Define $f:\mathbb{R}\rightarrow\mathbb{R}$ by $$f(x)=\begin{cases}0 & \text{ if }-\infty<x<0\\ 1 &\text{ if }0\leq x <1\\ x^{2}+x^{3} &\text{ if }1\leq x <2\\ 17 &\text{ if ...
3
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1answer
29 views

A strong version of the Dominated Convergence Theorem

Let $(X, \Sigma, \mu)$ be a measure space, and let $f, f_n:X\rightarrow \mathbb{C}$ be measurable functions with $f_n\rightarrow f$ pointwise. Assume that there are integrable functions $G, g_n:X\...
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1answer
26 views

$S$ be $\pi$-system on a set, given two measures on $\sigma(S)$, is there a topology on $\sigma(S)$ making $S$ dense, and the two measures continuous?

Let $\Omega$ be a non-empty set , $S \subseteq \mathcal P(\Omega)$ be a Pi system (https://en.wikipedia.org/wiki/Pi_system ) on $\Omega$ , let $\sigma(S)$ be the $\sigma$-algebra generated by $S$ (i.e....
9
votes
1answer
240 views

What is the cardinal of non measureable set?

we know that $|P( \Bbb{R} )|=|L (\Bbb{R} )|$ ( $L (\Bbb{R} ) $ is the set of all Lebesgue measureable set ) note that $L (\Bbb{R} ) \subsetneq P( \Bbb{R} ) $. What is the cardinal of non ...
2
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0answers
29 views

Random walk on d-dimensional torus

I am reading the following paper: http://arxiv.org/pdf/1602.03849v2.pdf I will explain the general setup below. Let $x\in X=\mathbb{T}^d$, where $\mathbb{T}^d$ is the d dimensional torus. Let $\rho$ ...
1
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1answer
36 views

Given a $\sigma$-algebra $\mathcal A$ on a set $X$, can we find a non trivial measure on $(X, \mathcal A)$?

Suppose that $X$ is a non-empty set, and $\mathcal A$ is a non-trivial $\sigma$-algebra on $X$. I was wondering whether it is possible to find a non trivial measure $m : \mathcal A \to \Bbb R_{≥0} \...
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2answers
39 views

Additive functions and measure theory

Key reference is the following: Hamel basis and additive functions Let's investigate real-valued functions $f(x)$ with the following (additive) property for all $\,a,b$ : $$ f(a+b)=f(a)+f(b) $$ It ...
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2answers
23 views

Sequence of integrable functions on $[0,1]$ with a certain bound on the $L_1$-norm converges to 0 a.e.

Given a sequence of integrable functions $f_n$ on $[0,1]$ that satisfy $\int_0^1 |f_n| dx \le 1/n^2$, I'm trying to show $f_n \to 0$ a.e. I was considering the set $A \subset [0,1]$ which consists of ...
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0answers
52 views

Self-learning measure theory for my background/need

I am not a math student, but I have taken courses in Calculus, Vector Algebra, Fourier and Laplace transforms, linear Algebra, elementary probability and stats while pursuing CompSci. I work as a ...
2
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1answer
24 views

Relation between pointwise convergence and convergence in measure.

Let $f_{n} \rightarrow f$ almost everywhere with $f$ integrable. Show that $\int |f_{n}-f|\rightarrow 0$ if and only if $\int |f_{n}|\rightarrow \int |f|$. Does this result still hold if we assume $f_{...
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2answers
71 views

Real Analysis, Folland Theorem 3.27 Properties of functions of Bounded Variation

Background Information: Taking $a = -\infty$ and considering the total variation as a function of $b$. To with $F:\mathbb{R}\rightarrow \mathbb{C}$ and $x\in\mathbb{R}$, we define $$T_F(x) = \sup\{\...
2
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1answer
37 views

Determining Class of a general Borel measure

Let $(X, \mathcal{T})$ be a topological space, and $\Sigma = \Sigma(\mathcal{T})$ the $\sigma$-algebra of Borel sets (that is, the $\sigma$-algebra generated by $\mathcal{T}$). In Real Analysis and ...
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2answers
90 views

Evaluating $\lim_{n\to\infty}\int_0^n(1-(x/n))^ne^{x/2}dx$

$$ \mbox{How to compute}\quad \lim_{n \to \infty}\,\,\int_{0}^{n}\left(1 -{x \over n}\right)^{n} \,\mathrm{e}^{x/2}\,\,\mathrm{d}x\,\,\, ?. $$ No ideas how to start this one. I see that the limit of ...
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0answers
16 views

Riesz measure in potential theory

I am studying Riesz measures associated to superharmonic funcions, following a book by Doob: Potential theory and its Probabilistic Counterpart. On page 51, the following theorem is introduced: If $u$...
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2answers
42 views

$\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin y}{ny(1+n^2y^2)}ndy$ via DCT?

I'm looking to calculate these limits/integrals: $$\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin (x/n)}{x(1+x^2)}dx$$ 2.$$\lim_{n\to\infty}\int_0^{\infty}\dfrac{\sin(x/n)}{(1+x/n)^n}dx$$ I posted ...
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1answer
22 views

$\pi$-system generating cylindrical $\sigma$-algebra

I have stumbled while solving the following problem. It seems simple, therefore your hints would be much valuable. Let $C$ denote the set of all continuos functions $x.$ from $t\in[0,\infty)$ to $\...
2
votes
1answer
34 views

Prove a set to be measurable set

Let $\left\{f_{n}\right\}$ be a sequence of measurable functions on $R$ and let $f$ be a measurable function on $R$.Prove that $\left\{x\in R| \lim_{n\rightarrow \infty}f_{n}\left(x\right)=f\left(x\...
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1answer
47 views

Compute $\lim_{p\to 1+}\left\|f\right\|_{p}$ where $f\in L^{1}[0,1] \cap L^{2} [0,1]$

Let $f\in L^{1}[0,1] \cap L^{2} [0,1]$. Compute $\lim_{p\to 1+}\left\|f\right\|_{p}$. I think the result would be $\left\|f\right\|_{1}$,but I don't know how to prove it.
2
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1answer
22 views

Showing that every $\sigma$-algebra is a semiring (of sets)?

Let $X$ be a set. A semiring (of sets) is a collection $\cal{S}\subset$ ${{\cal{P}}}(X)$ such that $$\emptyset\in\cal{S}$$ $$S,T\implies S\cap T\in\cal{S},$$and for $S,T\in\cal{S}$ there is a finite ...
4
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0answers
53 views

Inverse image is $\sigma$-algebra [on hold]

Let $(Y, \mathcal{A})$ be a measurable space and let $f$ map $X$ into $Y$, but do not assume that $f$ is one-to-one. Define $\mathcal{B} = \{f^{-1}(A) : A \in \mathcal{A}\}$. How do I see that $\...
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0answers
99 views

Proof that $\sum_{i=1}^\infty 2^i (\mathbf{1}_{A_i})^*(x)$ is finite almost surely [closed]

Let $(X, \mathcal{A}, m, T ) $ be a dynamical system and $f$ satisfying $\int |f| \ln^+ \ln^+ |f| {\rm d}m <\infty$. Put $A_i=\{x: 2^{i}\le f(x)< 2^{i+1}\}$ for $i\ge 2$ and $A_1= \{x:f(x)<4\...
1
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1answer
29 views

Why is compactness required for Brunn-Minkowski theorem?

Brunn-Minkowski theorem reads as follows: Consider two nonempty compact sets $A, B \subset \mathbb{R}^n$. Then the following inequality holds $$ [M(A+B)]^{\frac{1}{n}} \geq [M(A)]^{\frac{1}{n}} + [M(...
5
votes
1answer
30 views

Limit of measures, two questions on limits of integrals

Suppose $\mu_n$ is a sequence of measures on $(X, \mathcal{A})$ such that $\mu_n(X) = 1$ for all $n$ and $\mu_n(A)$ converges as $n \to \infty$ for each $A \in \mathcal{A}$. Call the limit $\mu(A)$. I ...
4
votes
0answers
36 views

Integration of Hilbert space valued mappings.

TL;DR: Is there a version of the Bochner integral which allows for the integration of isometric embeddings $\phi:X\to H$ from a metric space to a Hilbert space, satisfying $\int_X \|\phi\| d\mu < \...
2
votes
0answers
37 views

If $F$ is a closed subset of $[0, 1]$, then how do I see that there exists a finite measure on $[0, 1]$ whose support is $F$? [closed]

If $X$ is a metric space, $\mathcal{B}$ is the Borel $\sigma$-algebra, and $\mu$ is a measure on $(X, \mathcal{B})$, then the support of $\mu$ is the smallest closed set $F$ such that $\mu(F^\text{c}) ...
5
votes
1answer
47 views

What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?

Suppose $\epsilon \in (0, 1)$ and $m$ is Lebesgue measure. What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?