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

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5
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2answers
92 views

Number of equivalence classes of functions of real variable with the a.e relation.

What is the cardinal of the set $\mathcal{F}(\mathbb{R};X)/ \sim$ where $\sim$ is the relation $f\sim g \iff \mu(\{x\in \mathbb{R};f(x)\ne g(x)\})=0$ and $|X|=|\mathbb{R}|$? I guess that is ...
1
vote
1answer
69 views

Show that $\int_0^\infty \frac{\sin(x)}{x}e^{-xt}\,\mathrm{d}x=\frac{\pi}{2}-\arctan(t)$; $t>0$

I did this Let $I=\int_0^\infty\frac{\sin(x)}{x}e^{-xt} \,dx$ Then, $\frac{\partial I}{\partial t}=\frac{\partial}{\partial t} ...
2
votes
1answer
44 views

Cumulative distribution function implication

How can I prove the following: Let $X$ and $Y$ be two random variables. Suppose that their cumulative distribution functions satisfies $F_X(x)=F_Y(x)$ for all $x$. How can I show that $X$ and $Y$ are ...
1
vote
1answer
29 views

Pointwise limit of a measurable function is still measurable, for weak star convergence measure

Suppose I have a sequence of functions $(f_n(x))$: $\mathbb R^N\to\mathbb R^M$ such that $|f_n(x)|=1$ a.e. $x\in \mu_n$ where $\mu_n$ is a finite Radon measure over $\mathbb R^N$, and $f_n(x)$ is ...
0
votes
1answer
26 views

Does the following result require the random variables to be independent?

I am sitting with the book Labelled Markov Processes by Prakash Panangaden, and on page 79 he defines what it means for a set of random variables on a probability space to be independent, and after ...
5
votes
1answer
37 views

Existence of approximating simple function

Let $(X,\mathcal F,\mu)$ be measurable space with $\mu(X)<\infty$. $\mathcal F$ is $\sigma$-algebra on X and $\mathcal F$ is generated by algebra $\mathcal F_0$. Prove that for every measurable ...
4
votes
1answer
42 views

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

Let $(X,\mathcal F, \mu)$ be measurable space with $\mu(X)<\infty$. Prove that if function $f$ is measurable and finite on $X$ then $$\lim_{n\to \infty}\mu \{x: |f(x)|\geq n\}=0.$$ I have been ...
0
votes
1answer
28 views

Proof that a homeomorphic image of a non-borel set is non-borel

This question seeks to expand the proof given in the answer to this question. I am weak in topology, and am wondering if someone can provide a proof of why a homeomorphic image of a non-borel set is ...
1
vote
1answer
37 views

Mean value theorem for Lebesgue integral

Let $f$ be a mesurable function and $g$ be integrable function, and $\alpha, \beta$ are real numbers such that $\alpha \leq f \leq \beta$ a.e . Prove that there exists a real number $\gamma \in ...
2
votes
1answer
37 views

Help with a question about $\liminf \limits_{n \to \infty } \int_X {{f_n}\,d\mu } $ and $\int_X {\liminf \limits_{n \to \infty } {f_n}\,d\mu } $

The question is Let $(X, \cal M, \mu)$ be a finite measure space and fix $E\in \cal M$. For each $n \in \Bbb{N}$ define the function $f_n:(X, \cal M) \to \Bbb{R}$ given by $$ {f_n}(x) = \left\{ ...
1
vote
0answers
29 views

Expectation in measure theory

I'm reading a book on measure-theoretic probability, and the author defines the expectation of a random variable $X$ on a probability space $(\Omega,\scr H,\mathbb{P})$ as $\int_\Omega Xd\mathbb{P}$, ...
8
votes
2answers
106 views

Show that a given set has full measure or measure 0

Let $E \subseteq \mathbb R$ be Lebesgue measurable. And $E + q = E$ for any rational number $q$. Show that either $E$ or its complement has measure $0$. I tried this problem for few hours but ...
0
votes
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 ...
1
vote
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
votes
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
votes
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
votes
2answers
48 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 $(\Omega, F)$ ...
1
vote
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 ...
0
votes
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
votes
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 $(\Omega, F)$ ...
1
vote
2answers
70 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 ...
1
vote
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 ...
0
votes
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
votes
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
votes
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
vote
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
vote
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
votes
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 $3) A,B\in S, A\subset B \Rightarrow B\setminus A \in S$ ...
0
votes
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
votes
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
votes
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
vote
1answer
52 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
vote
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
votes
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
votes
1answer
49 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
vote
2answers
44 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
votes
2answers
50 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
votes
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
votes
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 ...
1
vote
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
votes
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 $X = Y = \mathbb{N}$, ...
0
votes
0answers
22 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
vote
1answer
34 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
vote
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
votes
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 ...
0
votes
0answers
27 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
votes
1answer
38 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 ...
-1
votes
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?