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

learn more… | top users | synonyms (1)

-1
votes
2answers
24 views

measure of a set which is a subset of infinitely many subsets of probability measure space

Let $B,A_1,A_2,....$ be the subsets of a probability measure space. If $ B \subset \bigcup A_j$, show that $m(B) \le \sum_{j=0}^\infty m(A_j)$. I have no idea as how to approach it. I do have the ...
10
votes
0answers
69 views

Is there a probability measure on $[0,1]$ with no subsets with measure $\frac{1}{2}$?

I have a decidedly weird question. Does there exist a probability measure $(\mu, \mathcal{F})$ on $[0,1]$ such that 1) $\mu(x) = 0$ for every $x \in [0,1]$ 2) For every $r \in [0,1] \setminus \...
3
votes
3answers
1k views

Convergence in measure of products

Let $\mu$ be a measure on $(X,\mathcal A)$ and let $f, f_1, f_2,\dots$ and $g, g_1, g_2,\dots $be real valued $\mathcal A$- measureable functions on $X$. Show that if $\mu$ is finite, $(f_n)$ ...
10
votes
1answer
1k views

Monotone class theorem

I have some question about the Monotone Class Theorem and its application. First I state the Theorem: Let $\mathcal{M}:=\{f_\alpha; \alpha \in J\}$ be a set of bounded functions, such that $f_\...
-2
votes
0answers
19 views

Limit in finite measure.

Is it true that if $(X, \mathcal{M}, m)$ be a finite measure space and $(E_{n} : n \in \mathbb{N})$ be a sequence of measurable set. Then $\lim_{n} m ( \bigcup_{k \geq n} E_{k})$ and $\lim_{n} m ( \...
0
votes
1answer
27 views

Integrals w.r.t. measure $A\mapsto \int_Agd\mu$

Let $\mu$ be a $\sigma$-additive complete measure* defined on the $\sigma$-algebra of the sets of unit $X$. If $g\in L^1(X,\mu)$ is a non-negative function, then, if I am not wrong, we can define a ...
4
votes
1answer
42 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 ...
41
votes
2answers
16k views

Limit of $L^p$ norm

Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, $\lim_{p\to\infty}\|f\|_p=\|f\|_\...
2
votes
2answers
52 views

information measure for matrix that is analogous to rank

Is there a measure for matrix that is analogous to rank of the matrix, but it is continuous on matrix elements? Say, we could say the information in identity matrix $I_n$ is $n$, and when the off-...
1
vote
0answers
28 views

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

I want to show that if $E$ is Jordan measurable then $m(A-B) \leq \epsilon$ where $A \subset E \subset B$. I think I have the right ideas but feel I am missing some details. I'd like some feedback ...
2
votes
1answer
47 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 ...
5
votes
1answer
49 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 ...
0
votes
0answers
4 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 $d$-...
7
votes
2answers
134 views

Intuitive Explanation of Why the Power Set of $\mathbb{R}$ is “too big” for the Lebesgue Measure?

I've been working with the construction of measures for a little bit, and I understand that in order for the Lebesgue measure to be an official measure on $\mathbb{R}$, we need to restrict it to a ...
0
votes
1answer
47 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 $\mathbb{R}$ ...
1
vote
0answers
21 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 ...
0
votes
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 $\...
1
vote
1answer
25 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 ...
14
votes
1answer
886 views

Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
1
vote
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 ...
1
vote
1answer
24 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 $\mathbb{R}^n$, ...
1
vote
2answers
56 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 ...
1
vote
1answer
45 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 ...
4
votes
1answer
67 views

What is the difference between $\mathbb E[Z|\mathcal G]=Y$ and $\mathbb E[Z|\mathcal G]\stackrel{\text{a.s.}}{=}Y$?

I'm somewhat confused by the definition of martingale: Let $(\Omega, \mathcal F, \mathcal F_n, \mathbb P)$ be a filtered probability space. We call $(X_n)_{n\in\mathbb N}$ martingale if for all $n\...
2
votes
1answer
24 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 ...
6
votes
1answer
96 views

Example of a set and monotone class where monotone class is not a $\sigma$-algebra?

What is an example of a set $X$ and a monotone class $\mathcal{M}$ consisting of subsets of $X$ such that $\emptyset \in \mathcal{M}$, $X \in \mathcal{M}$, but $\mathcal{M}$ is not a $\sigma$-algebra?
4
votes
0answers
96 views
+500

Limit of uniformly converging volume-preserving homeomorphisms

Definition A map $f\colon \mathbb{R}^n \to \mathbb{R}^n$ is volume-preserving if for every (Lebesgue) measurable set $V\subset \mathbb{R}^n$, $\mathcal{L}^n(V) = \mathcal{L}^n(f(V))$. I am wondering ...
1
vote
2answers
74 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\{\...
-1
votes
2answers
56 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\...
1
vote
0answers
43 views

Discrete random variable whose cdf is not a step function [on hold]

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 $F_{X} (x)$...
0
votes
1answer
387 views

Sub sigma algebra and probability spaces — definition

I am reading this book and I am a bit lost with the definitions because they are not provided and I can't seem to find it online: Let $L_2(\Omega,A,P)$ be a probability space such that $f \in L_2$ ...
0
votes
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 $A\subset \mathbb{R}^n$, ...
2
votes
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$...
0
votes
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)\} $ ...
-2
votes
0answers
33 views

Hölder inequality application to show that f=1

I want to proof that if $f \in L^{1}_{\mu}(\mathbb{R}), f > 0$ continuous, satisfies $(\int_\mathbb{R} f(x)d\mu)^{3} \le \int_\mathbb{R} f(x)^{3sin^{2}(x)}d\mu * (\int_\mathbb{R}f(x)^{\frac 32cos^{...
4
votes
1answer
32 views

Indicator function for limsup, liminf [duplicate]

If $A_i$ is a sequence of sets, define$$\liminf_i A_i = \bigcup_{j = 1}^\infty \bigcap_{i = j}^\infty A_i, \quad \limsup_i A_i = \bigcap_{j = 1}^\infty \bigcup_{i = j} A_i.$$Given a set $D$ define the ...
2
votes
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\{...
3
votes
1answer
5k views

Bounded Function Which is Not Riemann Integrable

This problem is taken from Problem 2.4.31 (page 84) from Problems in Mathematical Analysis: Integration by W. J. Kaczor, Wiesława J. Kaczor and Maria T. Nowak. Give an example of a bounded function ...
2
votes
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\...
3
votes
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
votes
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 $4$ ...
1
vote
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 ...
11
votes
3answers
2k views

What is the motivation of Levy-Prokhorov metric?

From Wikipedia Let $(M, d)$ be a metric space with its Borel sigma algebra $\mathcal{B} (M)$. Let $\mathcal{P} (M)$ denote the collection of all probability measures on the measurable space $(...
2
votes
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^{\...
3
votes
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\...
0
votes
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
245 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 ...
1
vote
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} \...
0
votes
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 ...
2
votes
0answers
30 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 $\rho$ ...