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

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1answer
31 views

The weak star convergence of Jordan decomposition

Given $\mu_n$ and $\mu$ finite signed Radon measures on the domain $\Omega$. We assume $\mu_n\to \mu$ in weak* sense, i.e. $\int_{\Omega}\phi \,d\mu_n \to \int_{\Omega} \phi\, d\mu$ for all test ...
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3answers
46 views

Show $\forall \epsilon > 0$ there exists $\delta > 0$ such that $\int_E {|f|d\mu } < \varepsilon $ for all $E\in \cal M$ with $\mu(E) < \delta$

The problem is Let $(X,\cal M, \mu)$ be a measure space and consider $f\in L^1(X,\cal M, \mu)$. Show that for each $\epsilon > 0$ there exists $\delta > 0$ such that $\int_E {|f|d\mu } < ...
0
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0answers
18 views

limiting and monotonic decreasing double sequence of probability measures

I am trying to figure out the behavior of this double sequence of measures. If I have a probability measure $\mu_n$ which is indexed by $n$, and a set of intervals $\mathcal{I}_k$ indexed by $k$ with ...
0
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1answer
39 views

Outer Measure exercise

This comes from an exercise from Real Analysis by Folland. Let $\mathcal{A}\subset P(X)$ be an algebra, $\mathcal{A}_\sigma$ the collection of countable unions of sets in $\mathcal{A}$, and ...
3
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2answers
49 views

What's the relationship between a measure space and a metric space?

Definition of Measurable Space: An ordered pair $(\Omega, \mathcal{F})$ is a measurable space if $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$. Definition of Measure: Let $(\Omega, F)$ ...
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1answer
33 views

If $f, g$ are measurable functions, then $f+g$ is measurable

Show that $f(x)+g(x)<a$ iff there exists rational number $r,q$ such that $r+q<a$ and $f(x)<r; g(x)<q$. Use this to prove if $f, g$ are measurable functions, then $f+g$ is ...
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2answers
31 views

if $\mu(X)$ is finite then $\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0$

Let $(X,\mathcal F, \mu)$ be measurable space with $\mu(X)<\infty$. a) Prove that if function $f$ is measurable on $X$ then $$\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0.$$ b) Can we ...
0
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1answer
42 views

What does Lebesgue measure space look like?

Definition of Measurable Space: An ordered pair $(\Omega, \mathcal{F})$ is a measurable space if $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$. Definition of Measure: Let $(\Omega, F)$ ...
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2answers
72 views

Null set squared is a null set

I'm attempting to find a solution to the following problem that doesn't involve splitting this into various cases. The question is: "If $m^*(E) =0$, show that $m^*(E^2) = 0$, where $E^2 = \{x^2 ...
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0answers
13 views

Do there exist equidistributed countable subgroups in (compact) Lie groups?

By an equidistributed countable subgroup I mean a countable subgroup (with a finite or possibly countable set of generators) that is dense in $G$ such that for any sufficiently nice function (Haar ...
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1answer
18 views

Permutation of a finite number of measurable functions is measurable?

Let there be a finite number of measurable functions $\{f_i\}_{i=1}^n$ with common domains of definition. Is it then true that a permutation of these functions $\{h_i\}_{i=1}^n$ also measurable? By ...
0
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1answer
44 views

Outer measure problem

This comes from an exercise from Real Analysis by Folland. Let $\mathcal{A}\subset P(X)$ be an algebra, $\mathcal{A}_\sigma$ the collection of countable unions of sets in $\mathcal{A}$, and ...
0
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0answers
33 views

Proof that if $f,g : X \to \bar{\mathbb{R}}$ are measurable then $f g$ is measurable.

Let $\bar{\mathbb{R}}$ be the extended real line (i.e. including $\{\pm \infty\}$). Proposition: If $f,g : X \to \bar{\mathbb{R}}$ are measurable then $f g$ is measurable (where $0 \cdot (\pm ...
1
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1answer
30 views

Exercise, show inequality(measure theory).

This exercise resembles what we do when we create the Lebesgue measure, but it is not quite the same. An interval can be any type: ...
1
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1answer
51 views

The outer measure on $X$ has a collection $M$ that is a $\sigma$-algebra

This is part of Caratheodory's Theorem taken by Real Analysis, Folland If $\mu^*$ is an outer measure on $X$, the collection $M$ of $\mu^*$-measurable sets is a $\sigma$-algebra. We first need to ...
2
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0answers
63 views

Extension of an additive function

Let $X$ be a finite set, $S\subset \mathcal P(X)$ such that: $1) X\in S$, $2) A,B\in S, A\cap B=\emptyset \Rightarrow A\sqcup B\in S$ and $3) A,B\in S, A\subset B \Rightarrow B\setminus A \in S$ ...
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1answer
21 views

prove that $\lim_{m \rightarrow \infty} \Sigma_{k=-m^2}^{m^2}|\int^{(k+1)/m}_{k/m}f(x)dx|=\|f\|_{L^1 (\Bbb R)}$.

Suppose $f \in L^1 (\Bbb R)$, prove that $$\lim_{m \rightarrow \infty} \sum_{k=-m^2}^{m^2}\left|\int^{(k+1)/m}_{k/m}f(x)\,dx\right|=\|f\|_{L^1 (\Bbb R)}.$$ For this one, it's easy to prove when $f$ ...
2
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1answer
31 views

Measure preserving transformation $T([a,b])\subset P$ if $\lambda(P)=\lambda([a,b])$

"Suppose that a measurable subset $P \subset [0,1]$ and the interval $I = [a,b] \subset [0,1]$ are such that $\lambda(P) = \lambda(I)$, where $\lambda$ is the Lebesgue measure on $[0,1]$. Show that ...
8
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1answer
107 views

Sufficient Condition for $f\in L^{1}(\mathbb{R}^{d})$ to belong to $L^{2}(\mathbb{R}^{d})$

Question. Let $\left\{\varphi_{j}\right\}$ be a complete orthonormal system for $L^{2}(\mathbb{R}^{d})$ such that each $\varphi_{j}\in C_{b}(\mathbb{R}^{d})$ (the space of continuous, bounded ...
1
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1answer
59 views

Integration with respect to a concrete measure

I got the problem of integrating with respect to a measure in concrete detail. Im just finding formal stuff elsewhere. The measure $Q(A)=\int_0^\infty P(f(r,X)\in A)dr$ is given and i need to show ...
1
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1answer
20 views

Do finite additivity and countable subadditivity imply countable additivity?

Given a Measure Space and f a positive set function on the sigma-algebra of the space (not identically infinite), how could I prove that f is a measure given the hypothesis above? I've tried both by ...
2
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1answer
50 views

Help with calculating a Riemann-Stieltjes integral.

The question is to calculate $\int_{ - 1}^2 {xd\omega (x)} = 0$ where $\omega (x) = \left\{ {\begin{array}{*{20}{c}} 0&{1 \le x \le 2}\\ 1&{0 \le x < 1}\\ 2&{ - 1 \le x < 0} ...
2
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1answer
50 views

Prove that $\int f(x)g(y) \,d(\mu \times \nu) = [\int f\, d\mu ][\int g \,d\nu]$

I got stuck on this problem to figure out how to calculate the integral on left-handed side, because we can't use Tonelli-Fubini theorem for this problem (lack of $\sigma$-finite condition). Hope ...
1
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2answers
45 views

A non-trivial example of $\cal F$- measurable function?

Given a measurable space $(\Omega, \cal{F})$, $f:(\Omega, \cal F) \to (\Bbb{B},\cal B)$, where $\cal B$ is the Borel $\sigma$-algebra of $\Bbb R$, is said to be $\cal {F}$-measurable if $f^{-1}(B)\in ...
3
votes
1answer
65 views

Multiplication operators on $L^2$

Let $X$ be a $\sigma$-finite measure space, and let $g$ a measurable complex-valued function $X$, which lies in $L^\infty(X)$. I would like to determine sufficient and necessary properties for the ...
0
votes
1answer
34 views

Do these two integrals agree?

Let $(S,\Sigma,\mu)$ be a measure space. Let $\tilde \Sigma$ be a $\sigma$ algebra on $S$ such that $\tilde \Sigma \subset \Sigma$. Then, $(S,\tilde \Sigma,\mu)$ is a measure space in its own right. ...
3
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2answers
51 views

Question about Kolmogorov extension theorem

I need some help understanding the relationship between the following two theorems Theorem 1: Let $\{\mu_n\}$ be a sequence of probability measures on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, where ...
4
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1answer
50 views

An mixed weak star convergence problem

Let $\Omega\subset \mathbb R^N$ open bounded. Given a sequence of Radon measure $(\mu_n)$ such that $\mu_n\to \mu$ in weak star sense in $\mathcal M_b(\Omega)$ and $\|\mu_n\|\nearrow \|\mu\|$. Also ...
2
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0answers
27 views

Caratheodory criterion, sufficient but is it necessary?

When constructing the Lebesgue measure on $\mathbb{R}$, it is shown that the sets that satisfy the caratheodory criterion form a sigma-algebra, and also that the countable additivity of the Lebesgue ...
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0answers
36 views

Proof - Limits of CDF

For a cdf, defined as $F(x)=P(X\le x)$, in order to prove $\lim\limits_{x\,\uparrow\,\infty}F(x)=1$, I've two concerns: (1) Some concern about a proof from a book, and (2)Validity of a proof that I've ...
3
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1answer
68 views

Calculate $\iint{f d\mu dv}$ and $\iint{f dv d\mu}$

The purpose of this problem is to show that in Fubini-Tonelli theorem, the condition $f \in L^{+}(X \times Y)$ or $f \in L^1$ is necessary. Here is the problem: Let $X = Y = \mathbb{N}$, ...
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0answers
23 views

Correspondence between AB-divergence and Kullback-Leibler divergence

I'm reading up on AB-divergence (alpha-beta-divergence) based mainly on the exposition given in Chichoki et al. (2011), "Generalized Alpha-Beta Divergences and Their Application to Robust Nonnegative ...
1
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1answer
36 views

Doubts regarding the upper bound for Total Variation

I was studying a chapter on Total Variation & Compactness, where I had gone through the following portion: " We can also relate the total variation with the shifted $L^{1}$-norm. Define: ...
1
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1answer
50 views

The approximation of supremum of a set.

Given $\Omega \subset \mathbb R^N $ be open and let $g$: $\Omega\to \mathbb R^+$ be a $l.s.c$ function such that $g\geq 1$, but not necessarily bounded above. Also assume that there exists a sequence ...
0
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0answers
29 views

How is $\mathcal F_\infty$ different from $\bigcup_{n=0}^\infty \mathcal F_n$? [duplicate]

Let $(X_n)_n$ be a sequence of random variables. Define $\mathcal F_\infty := \sigma(X_0, X_1, \ldots)$ and $\mathcal F_n := \sigma(X_0, X_1, \ldots, X_n)$. In the proof of the Kolmogorov's zero–one ...
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0answers
28 views

Sigma algebra and measure

I'm reading about Sigma-algebra on Wikipedia, but somehow can't really understand the motivation for its definition or why we need it. It is mentioned here in the 'Measure' section that we get rid of ...
2
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1answer
43 views

Distributions, PDFs, and Random Variables in Measure Theory

I'm currently reading a book on measure-theoretic probability theory, and I'm having trouble seeing how the familiar objects distributions, pdfs/pmfs, and random variables from my calc-based prob/stat ...
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4answers
47 views

On characteristic function

Let $X$ be a set and $A,B\subset X$. Can we consider $\mid\chi_A-\chi_B\mid$ as a characteristic function of some subset of $X$? If yes which subset?
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1answer
19 views

Question on Absolute Continuity of measures [closed]

Can I ask why does the question of absolute continuity of measure require the assumption of sigma-finiteness ? Thanks!
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2answers
39 views

Is this a measure on the sigma algebra of countable and cocountable subsets of R?

Consider the measurable space $(\mathbb{R}, \Sigma)$, where $$\Sigma := \{ A \subset \mathbb{R} \,:\, A \text{ is countable or } A^c \text{ is countable}\}.$$ Proving this is indeed a ...
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1answer
22 views

Comparing expectation values on different measures

I'm interested to know if it's possible to construct an inequality for the expectation values of a certain function over two different measures, i.e. can we say anything about these two equations? ...
3
votes
1answer
40 views

Show that $\mu(f)\mu(1/f)\geq\mu(\Omega)^2$

Prove that $\mu(\Omega)^2\leq\int f \,d\mu\int\frac{1}{f}\,d\mu$. I don't know if that what I did is correct or if it will help to solve the problem, but here it is: Using the Hölder inequality ...
0
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1answer
19 views

Direct Integral: Measurability

Given a Borel space $\Omega$. Consider plain functions: $$\eta,\vartheta\in\mathcal{F}(\Omega):=\{\eta:\Omega\to\mathbb{C}\}$$ The implication is wrong: ...
2
votes
1answer
49 views

Integral Measures: Identification

Problem Given a Borel space $\Omega$. Consider a Borel measure: $$\mu:\mathcal{B}(\Omega)\to\overline{\mathbb{R}}:\quad\mu\geq0$$ Regard a Borel measure: ...
2
votes
1answer
41 views

When is the image of a $\sigma$-algebra a $\sigma$-algebra?

Let $(E,\mathcal{E})$ and $(F,\mathcal{F})$ be measurable spaces and $f:E \rightarrow F$ with $f$ $\mathcal{E}/\mathcal{F}$ measureable. When is $f(\mathcal{E})$ a $\sigma$-algebra? I am aware that ...
0
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0answers
28 views

General step functions for lebesgue integral

For simplicity, I will only assume we are talking about the lebesgue integral on the same line. I read a construction of the riemann integral, that was designed in a way to resemble the construction ...
7
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1answer
108 views

Is there any $F \in \mathscr{F}$ such that $\mu(F)=x$?

Let $ (\Omega,\mathscr{F},\mu)$ be a probability space such that $\mu$ is non-atomic, and fix $x \in [0,1]$. Is it true that one can find $F \in \mathscr{F}$ for which $\mu(F)=x$? And what if $\mu$ is ...
0
votes
1answer
26 views

Show that $\left \{ \bigcup_{i\in I}A_{i}:I\subseteq \{1,\dots, n\} \right \}$ is a $\sigma$-algebra

Let $\{A_{i}\}_{i = 1}^{n}$ be a family of pairwise disjoint subsets of $X$. It is said that $$\mathcal{F}:=\left \{ \bigcup_{i\in I}A_{i}:I\subseteq \{1,\dots, n\} \right \}$$ is a $\sigma$-algebra. ...
3
votes
2answers
53 views

For what $p$ is $\frac{1}{(x(1+\ln(x)^2))^p}$ Lebesgue integrable?

I'm trying to use the fact that given $f:[a,\infty)\to\mathbb{R}$ Riemann integrable for every closed interval $[c,d]\subset [a,\infty)$, then $f$ is Lebesgue integrable if, and only if, ...
4
votes
1answer
20 views

Weak $L^p$ implies strong $L^q$ for $q<p$

Another prelim question... Suppose $0<q<p<\infty$, and $E\subseteq \mathbb{R}^n$ has finite measure. Suppose $f$ is in weak $L^p$, i.e. $\lambda(|f| > t) \leq N/t^p$. Show $f \in L^q(E)$ ...