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

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2
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1answer
199 views

Question regarding Lebesgue Integrability in $\sigma$ -finite spaces

I'm taking a course in measure theory and we defined integrability in a $\sigma$ -finite space as follows: Suppose $\left(X,\mathcal{F},\mu\right)$ is a $\sigma$-finite measure space, a measurable ...
2
votes
1answer
97 views

Show that $f_n1_{A_n}$ convergences in mean

Consider the measurable space $(\Omega,\mathcal{A},\mu)$. Let $f,f_1,f_2,\ldots$ be measurable functions on that measurable space and $A,A_1,A_2,\ldots\in\mathcal{A}$. Let $(f_n)$ converge in ...
2
votes
0answers
121 views

Criterion for homeomorphisms to be absolutely continuous

Let $f : (0,1) \to \mathbb{R}$ be a continuous injective function (i.e., a homeomorphism onto its image). As such, $f$ has an inverse, and $f$ is differentiable almost everywhere. Let $E_f = \{x \mid ...
2
votes
0answers
66 views

$L^1$ approximate continuity points of epigraph of a continuous function

Consider a continuous function $f\colon \mathbb R\to \mathbb R$. Let $F(x,y):=\mathrm{sign\,}(y-f(x))$. Is it true that for a.e. $x\in \mathbb R$ the point $(x,f(x))$ is not a point of $L^1$ ...
2
votes
2answers
95 views

Function of a set of r.v.'s measurable w.r.t. the $\sigma$-algebra generated by the r.v.'s

Let $X_1, \dots, X_n$ be a set of random variables defined on a probability space $(\Omega, \mathcal{F}, P)$ and denote by $\mathcal{S} = \sigma(X_1, \dots, X_n)$ the $\sigma$-algebra generated by the ...
2
votes
0answers
75 views

Dominated Convergence on risk measures

This is a quite specific question and I am not able to provide the whole background (e.g. what a risk measure is). If someone knows that would be great. I am having difficulties understanding a ...
2
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0answers
67 views

Integrability and product measure

Let $X$ and $Y$ be subsets of $\mathbb{R}$, and let $\mu$ be a measure on $X$ and $\nu$ a measure on $Y$. Let $f : X \times Y \rightarrow \mathbb{R}$ be $\mu$-summable and $\nu$-summable, i.e. ...
2
votes
1answer
321 views

invariant measure under irrational rotation on $S^1$

Prove that if $T:S^1 \to S^1$ is an irrational rotation, then the only probability measure on $S^1$ that is $T-$invariant is the lebesgue measure or a multiple or it. We are considering the ...
2
votes
0answers
156 views

Lebesgue integral for a non-negative, measurable and bounded function

Consider the measurable space $(\Omega,\mathcal{A},\mu)$. Let $f$ be a measurable, non-negative and bounded function. Show that the $\mu$-integral of f is given by $$ \int f\, ...
2
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0answers
69 views

Estimate Borel Sets with Open Sets

We're given the measure space $(X,\mathscr{A}, \mu)$with $X=\bigcup_{i=1}^\infty X_i$with $X_i\subset X_{i+1}\subset ...$, $X_i$are open for all i and $\mu(X_i)<+\infty\space\space\space\space ...
2
votes
1answer
77 views

Help with Homework Problem

Let $f_k(x)=|x-1/k|^{-2/3} (k=1,2,3,...).$ Do the $f_k$ have an integrable majorant, meaning a function bounding $f_k$ that satisfies the dominated convergence theorem, on the interval [0,1] with ...
2
votes
1answer
44 views

Why this convergence should be uniform?

Let $f$ be an extended real valued measurable function. Then we need to show that there exists a sequence of real-valued simple functions that converge to $f$. We also need to show that if $f$ is ...
2
votes
0answers
134 views

Problem concerning Lebesgue Integral.

I am looking for feedback/corrections on the following solution attempt. I have only typed up one direction of the inequality; if this direction is correct, then I am certain that the other direction ...
2
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0answers
151 views

Each bounded measurable function $f:[a,b]\to\mathbb{R}$ is almost a Borel function.

The exact problem statement is: There exists a Borel function $h:[a,b]\to\mathbb{R}$ and a Borel set $H\subset[a,b]$ such that $f=h$ on $H$ and $m([a,b]\setminus H)=0$. My attempt of the proof is as ...
2
votes
0answers
97 views

Non-measurable function which is Riemann-integrable.

Is there a non-measurable, bounded function $f : [a,b] \to \mathbb{R}$ that is Riemann-integrable on $[a,b]$ ?
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0answers
73 views

Is $\sigma$-finiteness necessary for the existence and uniqueness of product measure?

Let $(X,\mathfrak{B}_X,\mu_X)$ and $(Y,\mathfrak{B}_Y,\mu_Y)$ be $\sigma$-finite measure spaces. Then there exists a unique measure $\mu_X\times\mu_Y$ on $\mathfrak{B}_X\times\mathfrak{B}_Y$ that ...
2
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0answers
65 views

Radon measure gives a monotone function

I am starting study on Lebesgue-Stieltjes measure. One of the problem that stuck me is: If $\mu$ is a Radon measure on $R$, show that there exists a monotone function $F: R \to R$ such that $\mu = ...
2
votes
1answer
55 views

Random variables tending to 0 a.s. but with $\mathbb{E}(sup_n|X_n|) = \infty$

I am trying to find a uniformly integrable sequence of random variables $(X_n: n \in \mathbb{N})$ such hat both $X_n \to 0$ almost surely and $\mathbb{E}(sup_n|X_n|) = \infty$. I think this is ...
2
votes
0answers
110 views

CDF of non-atomic singular measure

Suppose $\mu$ is a non-atomic measure on the Borel subsets of $[0, 1]$ such that $\mu$ and Lebesgue measure are mutually singular. Show that if $F$ is the cumulative distribution function of $\mu$, ...
2
votes
1answer
80 views

Intrinsically Ergodic Factor

Let $(X,T)$ and $(Y,S)$ be two intrinsically ergodic system with the same topological entropy i.e. $\exists ! \mu, \exists ! \nu$ measures of maximal entropy such that ...
2
votes
0answers
39 views

Weak convergence of measures for Cantor function

For positive integer $k$, let $\mu_k=\dfrac{1}{2}\left(\delta(x)+\delta\left(x-\dfrac{2}{3^k}\right)\right)$. Let $dC_k=\mu_1\ast\cdots\ast\mu_k$. In this question, it was shown that ...
2
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0answers
40 views

On $\mathbb{R}$ with the Borel $\sigma$-algebra, is $e^x$ measuable?

Well for any $\lambda \in \mathbb{R}$ we want that the set $S = \{x \mid f(x) > \lambda\}$ is in the Borel $\sigma$-algebra, $B$, i.e. the algebra generated by the intervals. Well firstly, for any ...
2
votes
1answer
240 views

Is a regular Borel measure on a locally compact space necessarily $\sigma$-finite?

I am trying to compile a proof of the uniqueness of Haar measure. Usually this is done by multiple-integral mumbo-jumbo, abusing left and right invariance of two potential measures and invoking ...
2
votes
1answer
85 views

Lipschitz-continuity and measurability

Problem: Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be Lipschitz-continuous with constant $L$. Let $\lambda_n$ be the $n$-dimensional Lebesgue-measure. Show that there exists a constant $C\ge 0$, s.t. ...
2
votes
0answers
65 views

Convergence a.s. implies convergence in $L^{1}$

Suppose $\lim_{n \to \infty}X_{n} = X$ almost surely. Let $Y=\sup _{n}|X_{n}-X|$. Show that $Y < \infty$ almost surely, and define a new probability measure $Q$ by ...
2
votes
0answers
33 views

Measure via circles?

In $\mathbb{R}^n$, if we preset the measure $\mu D$ of a (open or closed or whatever on the boundary) disk $D$, and use them to approximate sets, instead of squares $$m^*(X)=\inf_{\cup D_i\supset ...
2
votes
1answer
138 views

$A\cap(A+x) \neq \emptyset$ for a set $A$ with positive Lebesgue measure and $0<|x| < \delta$

can someone help me show that if $A$ is a measurable set with positive Lebesgue measure then there exists some $\delta >0$ such that $A\cap (A+x) \neq \emptyset$ whenever $|x|< \delta$? I ...
2
votes
1answer
84 views

measurability of metric space valued functions

Let's say that we have a measure space $(X, \Sigma)$ and a metric space $(Y, d)$ with its Borel sigma algebra. If $f_n: X\rightarrow Y$ is an arbitrary sequence of measurable functions, then I ...
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0answers
90 views

Does showing a function is integrable suffice to show the function is measurable?

I am reviewing past homework exercises in preparation for a midterm exam. Fortunately, my professor provides solutions. However, I found one of his solutions contains an (seemingly) important ...
2
votes
0answers
164 views

Problem involving Outer Measure

Below is my attempt at the solution of a problem involving outer measure. $\textbf{Problem:}$ For sets $A$ and $B$, prove that if $m^{*}(A)=0$, then $m^{*}(A \cup B)= m^{*}(B).$ ...
2
votes
0answers
42 views

Singular measure with respect to translates

Let $\mu$ be some Borel measure on $\mathbf{R}$ such that, for every $t \neq 0$, the push-forward $(\tau_t)_* \mu$ is singular with respect to $\mu$ (where $\tau_t(x)=x+t$). What can we say about ...
2
votes
1answer
145 views

Find an example of a sequence $\{f_k\}$ such that $f_k\in L^p$ for $1\le p <\infty$, $f_k\to0$ in $L^p$ for $1\le p < p_0$

Let $1<p_0<\infty$. Find an example of a sequence $\{f_k\}$ such that $f_k\in L^p$ for $1\le p <\infty$, $f_k\to0$ in $L^p$ for $1\le p < p_0$, but $f_k$ does not converge in $L^{p_0}$. ...
2
votes
0answers
40 views

show that $\lim_{n\to\infty} \mu^*(\sum_{k=1}^n A_k) = \mu^*(\sum_{k=1}^\infty A_k)$ for outer measure $\mu^*$

Let $\mu$ be a pre-measure on a ring $\mathcal R\subset\mathcal > P(\Omega)$ and $\mu^*$ the outer-measure induced by $\mu$ on $\mathcal > P(\Omega)$. Show that for sets ...
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0answers
275 views

Prove Lebesgue dominated convergence theorem directly

We can prove that Lebesgue dominated convergence theorem, monotone convergence theorem and Fatou's lemma are equivalent. Almost all the textbooks prove the monotone convergence theorem or Fatou's ...
2
votes
0answers
64 views

Integral of continuous function over probability distribution

Let $\mu_f$ denote the probability distribution of $f$ with $f(x)=10x-1$ for $x\in(0,1/2]$ and $f(x)=1$ for $x\in[1/2,1]$. If $g:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function, what is ...
2
votes
0answers
52 views

Example of when $\mathcal{B}(X\times Y) \neq \mathcal{B}(X) \times\mathcal{B}(Y)$ but $|X|,|Y| \leq |\mathbb{R}|$

I am interested in knowing examples of when $\mathcal{B}(X\times Y) \neq \mathcal{B}(X) \times\mathcal{B}(Y)$. By allowing $|X|,|Y|$ to be large we can provide a trivial counterexample, as in the one ...
2
votes
0answers
118 views

Integral Using Dominated Convergence Theorem

Prove that for $a>-1$ we have $\int_{[0,1]}x^{a}(1-x)^{-1}\log(x)\, dx=\sum_{n=1}^{\infty}\frac{1}{(a+n)^{2}}$. --It is shown previously that $\sum_{n=1}^{\infty}f_{n}(x)$ converges a.e. to an ...
2
votes
0answers
220 views

Borel Measure is Outer Regular iff it is inner regular?

I have a doubt on whether this theorem is true: Let $(X,\Gamma)$ be a compact space. Then a Borel measure $\mu$ on $\mathbb{B}(X)$ is outer regular iff it is inner regular. Can anyone shed some light ...
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0answers
66 views

Uniform convergence in measure space

Suppose $\mu(X)<\infty$. Suppose $f_n$ and $f$ are real-valued measurable functions on $X$ such that $f_n\rightarrow f$ almost everywhere. Prove that for every $\epsilon>0$, there exists ...
2
votes
1answer
69 views

invariant measure on a quotient of a topological group

Suppose I have a locally compact topological group, $G$, and a closed subgroup $H\leq G$. Suppose $\Delta _G|_H = \Delta_H$ where the $\Delta$ are the modular functions on $G$ and $H$. How can I see ...
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votes
0answers
43 views

On the density of this set in L^2[0,1]

This is part of an example in Rudin's Functional Analysis, in the chapter on unbounded operators. It needs to be shown that the set of absolutely continuous functions $f$ on $[0,1]$ with square ...
2
votes
1answer
41 views

Is the product sigma algebra generated by P(R) and P(R) equal to P(R^2)?

Is $P(\mathbb R) \otimes P(\mathbb R) = P(\mathbb R^2)$? I'm using the tensor notation $\otimes$ to denote the sigma algebra generated by sets of the form {$A \times B$: A, B $\subseteq \mathbb R$}. ...
2
votes
1answer
45 views

Finding $E(X^r\mid Y)$ of an exponential function

Let $(X,Y)$ denote a two-dimensional random vector with an absolutely continuous distribution with density function $$p(x,y) = \frac{1}{y}\exp(-y), \qquad 0 < x < y < \infty.$$ Find ...
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votes
0answers
92 views

For any $x∈E$,there exist an open ball $B(x,\epsilon_x)$ S.T $m_\star(E\cap{B(x,\epsilon_x))}=0$.Prove $m_\star(E)=0$.

Given $E\subset{R}^n$,for any $x∈E$,there exist an open ball $B(x,\epsilon_x)$ such that $$m_\star(E\cap{B(x,\epsilon_x))}=0.$$Show that $m_\star(E)=0$. My proof: Since $R^n$ is second-countable and ...
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0answers
1k views

Prove Reverse Fatou's lemma

I am trying to prove the reverse Fatou's lemma but I can't seem to get it. Here is my approach: We have a sequence $\lbrace f_k \rbrace$ in $\mathbb R_e$ and $E\subset \mathbb R^n$. We know that ...
2
votes
0answers
179 views

Completeness of metric space induced by outer measure (similar to Nikodym metric)

Let $S_\mu$ be a semi-ring of subsets of $X$ and $\mu$ be a $\sigma$-additive measure on $S_\mu$. Let $\mu^*$ be the induced outer measure on $P(X)$. Define a relation $\sim$ on $P(X)$ by ...
2
votes
0answers
48 views

Convergence of integral under a limit

Background: Let $X$ and $Y$ be two lognormal random variables, and $Z = X|Y = y$ a lognormal random variable obtained by conditioning on $Y$. Denote by $g_{\rho}(z)$ the probability density function ...
2
votes
0answers
101 views

A problem concerning measures on locally compact spaces

I am stuck on a question for quite sometime now, although in the text (http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/Jewett.pdf , Pg. 10, 2.3E ) it is said ...
2
votes
0answers
73 views

Proving product measures: Homework

Being an engineer I'm at loss how to prove the following exercises, and I would appreciate any comments. Prove Fubini's theorem for an $\mathcal{L}^1$ integrable function $f$. Here's my stab at it. ...
2
votes
0answers
46 views

Scheffe's theorem

Scheffe's theorem states that consider the real line $\mathbb{R}$ and sequence of probability densities $\{f_n\}$ such that $f_n \to f$ pointwise. Then if $\{P_n\}$ and $P$ denote the reps. measures ...