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

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1answer
161 views

Show that the difference of two Borel sets is itself a Borel set

I am currently in a Measure Theory class and we are going over Borel sets. I am having difficulty with the following proof: show that for any two Borel sets $A,B$, the difference $A-B$ is a Borel ...
5
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3answers
481 views

if convolution of $f$ with itself remains same, then $f=0$ a.e?

I'm trying to answer the question above.. But I'm not certain in either way. I tried to prove it by giving counter examples.. But it always failed.. Then i also tried to draw contradictions But ...
2
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1answer
73 views

Sequence in measure

I need to prove the following: If $\alpha$ is a probability measure and ${X_n \to X} $ a.e then ${X_n \to X}$ in measure. Show that the opposite may not be true.
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1answer
583 views

Properties of Borel sets and Lebesgue measure

Let $A$ be a set, let $B$ be a Borel set such that $B \subseteq A$. Because $B$ is a Borel set, can I automatically say that I can represent it as a countable union of closed sets. Thus I can ...
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1answer
82 views

Measure theory, integration.

Let $(X,S,\mu)$ be a measure space, and let $f,f_1,f_2,\dots:X\to [0,+\infty]$ be $\mu$-integrable such that $\lim\limits_{n\to\infty}f_n=f$ almost everywhere. Show that: ...
2
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2answers
186 views

The set related to the Cantor set

I am considering the set related to the Cantor ternary set.Let A be the set of numbers in [0,1] whose ternary expansions have only finitely many 1's.Prove that $\lambda(A)=0$ where $\lambda$ is the ...
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1answer
720 views

Reconciling several different definitions of Radon measures

Upon reviewing some basic real analysis I have encountered two different definitions for Radon measure. Let the underlying space $X$ be locally compact and Hausdorff. Folland's Real Analysis gives the ...
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1answer
100 views

a simple measure theory question (from homework)

Let X be a positive random variable independent of a standard Brownian motion B. Let $M_t = B_{tX}$ for t > 0. We assume that the random variable X is $F_t$ measurable for all t $\geq$ 0, require to ...
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0answers
55 views

Extension of complex measure

How answer this question? Could be a hint! Let $(X,M,\rho)$ be a finite measure space. Suppose $U \subset M$ is an algebra of sets and $\mu: U \longrightarrow \mathrm{C}$ is a complex, finitely ...
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1answer
250 views

Does the Riemann integral come from a measure?

Can we approach the Riemann integral with measure theory? That is: can we find a measure $\mu$ defined on a $\sigma$-algebra of $\mathbb{R}$ such that a function is $\mu$-integrable if and only if ...
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1answer
88 views

Comparing $\sigma$-algebras generated by 2 functions

I'm trying to solve the following exercise: Let $f,g:\mathbb R_+\rightarrow \mathbb R_+$ be given by $f(x) = \sum_{n=0}^\infty n\mathbb 1_{[2n,2n+2)}(x)$ and $g(x) = \sum_{n=0}^\infty\mathbb ...
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3answers
100 views

Proof of $\sigma(A^n)\supset\sigma\bigg(\sigma (A)^n\bigg)$?

Let ${\mathcal B}_n$ be the Borel $\sigma$-algebra on ${\mathbb R}^n$. Then it's not hard to show that $$ {\mathcal B}_n=\sigma(A^n) $$ where $$ A=\{(-\infty,a]: a\in{\mathbb Q}\}. $$ Let ${\mathcal ...
4
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0answers
172 views

Haar Measure: Unimodular Locally Compact Groups

I have the following problem: "Let $G$ be a locally compact group, all of whose normal subgroups are contained in $Z(G)$. Prove that $G$ is unimodular." My attempt at attacking the problem was to ...
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2answers
60 views

Show that $f_n$ doesn't converge to $0$ in $L^p$ where $f_n = n^{-1/p}\chi_{[0,1]}$

Let $\mathfrak{B}$ be the Borel $\sigma$-algebra on $\mathbb{R}$ and $m$ be the Lebesgue measure; let $1 \leq p < \infty$. Show that $n^{-1/p} \chi_{[0,1]}$ does not converge to $0$ in $L^p$. As ...
7
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1answer
391 views

Basics of Haar measure

Suppose $G$ is a locally compact group. Then $G$ has a left-invariant measure $dg$, say, which means that $$\int f (hg) dg = \int f(g) fg$$ for any test function integrable on $G$. The ...
6
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0answers
57 views

Shift operator on locally compact groups

Assume $f:G\rightarrow H$ is a measurable function between two locally compact abelian groups and let $T^h(f) = f\circ T^h$, where $T^h(x) = x-h$ (group operations in G and H are written additively). ...
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2answers
244 views

Paradox as to Measure of Countable Dense Subsets?

Consider the set $E=\mathbb{Q}\cap[0,1]$, and let $\{q_{j}\}_{j=1}^{\infty}$ be some enumeration of this countable set. For every $\epsilon>0$, the cubes $\{Q_{j}\}_{j=1}^{\infty}$ of length ...
2
votes
1answer
112 views

Derivating $f(t)=\int_0^t x dx$ using measure theory

For the function $f(t)=\int_0^t x dx$, Riemann fundamental theorem of calculus says $$ f'(t)=t $$ On Lebesgue side, I know a theorem says $$ f'(t)=t \text{, on } \mathbb R \setminus E \text{ where } ...
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1answer
208 views

The distribution of the sum of infinite fair coin tosses

This question came up in a course on measure theoretic probability theory. We have had lots of information on the existence of distribution, but no examples of how to find/construct them. Here's the ...
5
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0answers
1k views

Closure, Interior, and Boundary of Jordan Measurable Sets.

This question has a number of parts. Let $E\subset\mathbb{R}^{d}$ be a bounded subset. (1) Show that $m^{\star,(J)}(E)=m^{\star,(J)}(\bar{E})$ (closure) (2) Show that ...
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1answer
249 views

Prove that in $\mathbb{R}$, if $|a-b|>\alpha$ for all $a\in A$ and $b\in B$, then outer measure $m^*(A\cup B)=m^*(A)\cup m^*(B)$

Prove that for sets $A,B$ bounded in $\mathbb{R}$: If there exists $\alpha > 0$ such that $|a-b|>\alpha$ for all $a\in A$ and $b\in B$, then outer measure $m^*(A\cup B)=m^*(A)\cup m^*(B)$. So ...
1
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1answer
346 views

Looking for a non-measurable real-valued function with a specific property.

I would like to show that the measurability of the absolute value of a function $g:\mathbb{R} \rightarrow \mathbb{R}$ does not imply that $g$ is measurable (I am convinced that $g$ does not have to be ...
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0answers
72 views

What is a product measure space for the infinite case? [duplicate]

Possible Duplicate: Infinite product of measurable spaces Given measure spaces $(X_i, \Sigma_i, \mu_i)$ where $i$ ranges over some arbitrary index set, my understanding is that there has ...
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4answers
340 views

Is $A$ a Borel set?

Let be $X$ a metric compact space and $(G,+)$ a topological compact abelian group. Let be $\mathcal{A}$ the Borel $\sigma$-algebra of $X$ and $\mathcal{B}$ the Borel $\sigma$-algebra of $G$. ...
7
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1answer
1k views

Hölder inequality from Jensen inequality

I'm taking a course in Analysis in which the following exercise was given. Exercise Let $(\Omega, \mathcal{F}, \mu)$ be a probability space. Let $f\ge 0$ be a measurable function. Using Jensen's ...
5
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1answer
1k views

Set of points of continuity are $G_{\delta}$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. Show that the points at which $f$ is continuous is a $G_{\delta}$ set. $$A_n = \{ x \in \mathbb{R} | x \in B(x,r) \text{ open }, ...
2
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1answer
277 views

Is the pull back of a generated $\sigma$-algebra itself a generated $\sigma$-algebra?

Let $f : \Omega_{1} \rightarrow \Omega_{2}$ be a map from a measurable space $\Omega_{1}$ to another measurable spacce $(\Omega_{2},\mathcal{B})$, where $\mathcal{B}$ is the generated $\sigma$-algebra ...
3
votes
3answers
3k views

Generate the smallest $\sigma$-algebra containing a given family of sets

My teacher gave me an example of performing the subject: Example Let $\Omega = \Bbb R$ and $\mathcal R = \{(-\infty,-1),(1,+\infty)\}$. Then $\sigma(\mathcal R) = \{\emptyset, \Bbb R, ...
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2answers
69 views

$u\in\mathcal{L^{1}(\mu)}$ (integrable) and $v\in\mathcal{L^{1}(\mu)}\Rightarrow u\cdot v\in\mathcal{L^{1}(\mu)}$

Let $(X,\mathcal{A},\mu )$ be some measure space. If $u\in\mathcal{L^{1}(\mu)}$ (integrable) and $v\in\mathcal{L^{1}(\mu)}$ (integrable) can I then deduce that $u\cdot v\in\mathcal{L^{1}(\mu)}$? ...
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1answer
460 views

Example of a decomposable measure space that is not sigma finite

A measure space $(X,\mathfrak{M},\mu)$ is decomposable if $X$ is a disjoint union of measurable subsets, $X=\bigcup_{i\in I}X_{i}$, with $\mu(X_{i})<\infty$ for all $i$, and $\mu(A)=\sum_{i\in ...
2
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2answers
200 views

$E$ is measurable, $m(E)< \infty$, and $f(x)=m[(E+x)\bigcap E]$ for all $x \in \mathbb{R}$

Question: $E$ is measurable, $m(E)< \infty$, and $f(x)=m[(E+x)\bigcap E]$ for all > $x \in \mathbb{R}$. Prove $\lim_{x \rightarrow \infty} f(x)=0$. First, since measure is translation ...
2
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0answers
227 views

conditional expectation and order statistic

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space.Let $\ X=(X_1,..,X_n)$ a random vector, with$\ n$ independents random variables whose law is $\mu$ on $\mathbb{R}$. We define ...
2
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1answer
215 views

Is there a measure

Is there a measure $\nu$ on $[0,\infty)$ such that $$ \ln x=\int_{0}^{\infty}d\nu\left(y\right)/\left(x+y-1\right)? $$ Thanks for any helpful answers!
5
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2answers
821 views

Prove Borel sigma-algebra translation invariant

Can anyone explain: Let $B$ be a Borel set and $B + a = \{ x + a : x \in B\}$. Why is $B + a$ a Borel set? I think I have to use some good set principle but not sure how to complete the proof.
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1answer
126 views

invertible, measurable and measure preserving

$T: [0,1)^{2}\rightarrow[0,1)^{2}$ by $T(x,y) = (2x,\frac{y}{2})$, with $0 \leq x < \frac{1}{2}$ and $T(x,y) = (2x-1, \frac{y+1}{2})$, with $\frac{1}{2} \leq x < 1$ In class we said this $T$ ...
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2answers
2k views

How should I understand the $\sigma$-algebra in Kolmogorov's zero-one law?

I'm learning Kolmogorov's zero-one law in probability theory: Let $(Ω,{\mathcal F},P)$ be a probability space and let $F_n$ be a sequence of mutually independent $\sigma$-algebras contained in ...
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1answer
845 views

Almost Sure Convergence Using Borel-Cantelli

I am working on the following problem: Let $(f_n)$ be a sequence of measurable real-valued functions on $\mathbb{R}$. Prove that there exist constants $c_n > 0$ such that the series $\sum c_n ...
7
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3answers
379 views

Extension of the Lebesgue measurable sets

My question is the following : is there a $\sigma$-algebra $\mathcal{T}$ (of subsets of $\mathbb{R^n}$) that contains strictly the $\sigma$-algebra $\mathcal{L}$ of Lebesgue measurable sets (in ...
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1answer
243 views

Lebesgue measuarable sets under a differentiable bijection

Let $U,V \subseteq \mathbb{R}^{n}$ be open and suppose $A\subseteq U$ are (Lebesgue) measurable. Suppose $\sigma \in C^{1} (U,V)$ be a bijective differentiable function. Then does it follow that ...
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0answers
69 views

A measure realated to Riemann integral

Let $( \mathbb{R}^k , \mathcal{A} , m_{k} )$ be a Lebesgue measurable space, i.e., $m_{k}=m$ is a Lebesgue measure. Let $f: \mathbb{R^k} \to \mathbb{R}$ be a $m$-integrable function. Define a function ...
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1answer
107 views

What can we say about the image of a measureable map in the support of its push forward measure

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and $f:\Omega\rightarrow\mathbb{R}^d$ a measurable function. Let $\mu$ be the probability measure defined by $\mu(B):=\mathbb P(f^{-1}(B))$ ...
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2answers
561 views

Riemann-Lebesgue Integrable

Show that there is no riemann integrable function $f$ on $[0,1]$ such that $f=\chi_{C}$ a.e. (almost everywhere), where $C$ is the fat cantor set. Proof: Would it suffice to show that $\chi_C$ is ...
9
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1answer
2k views

If $E$ is Lebesgue measurable, show that there exists a closed set $F$ with $F \subset E$ and $m(E\setminus F)<\epsilon$

Just having trouble with this problem. First, it says to prove that if $E$ is Lebesgue Measurable, and $\epsilon>0$ is arbitrary, then there is an open $O$ such that $E \subset O$ and $m(O\setminus ...
1
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2answers
250 views

Measure and Outer Measure Definition

I would like to find exemples to show and demonstrate that each of the statements of the definition of: -measure $\mu\left(\emptyset \right)=0$ $\mu \left( \bigcup A_n\right)=\sum \mu \left( ...
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2answers
2k views

Closed under countable union

I am reading a tutorial on measure theory and it states: "Given an interval $E = [a, b]$ and a set $S$ of subsets of $E$ which is closed under countable unions, we define the following..." I was ...
1
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1answer
164 views

Probability measure with predefined support

Let $I$ be an index set, possibly uncountable, and $U_\iota\subseteq\mathbb{R}^d$ be closed for $\iota\in I$. Does there always exist a Borel probability measure $\mu$ such that ...
0
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0answers
269 views

Jacobian with Chain rules

$$S\subseteq \mathbb{R}^N, U\subseteq \mathbb{R}^k, \phi\in C^1(U,\mathbb{R}^N) \ \text{ s.t.} \ \phi(U)=S.$$ $J_\psi$ is the Jacobian matrix. Let $V\subseteq\mathbb{R}^k$ be open, and let $\psi:V\to ...
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2answers
82 views

Prove $ \int_{cX} \frac{dt}{t} = \int_{X} \frac{dt}{t}$ for every Lebesgue measurable set $X$

Let $c>0$. Let $X \subseteq (0,\infty)$ be a Lebesgue measurable set. Define $$ cX := \{ cx \mid x \in X \}. $$ Then $$ \int_{cX} \frac{dt}{t} = \int_{X} \frac{dt}{t}$$ Now I can prove this for ...
2
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0answers
107 views

Has this function transformation on $L^1(\mathbb{R})\;$ been named and researched?

I'm sorry, but my measure theory is practically non-existent. I'm looking at this transformation: $$T : L^1(\mathbb{R}) \rightarrow \mathbb{R}^\mathbb{R}$$ $$\begin{aligned} (Tf)(x)&:=\mu\{ y: |y ...
2
votes
1answer
343 views

Lipschitz functions carry $F_\sigma$ to $F_\sigma$.

Let $f:[a,b] \rightarrow \mathbb{R}$ be a Lipschitz function. I want to show that it carries $F_\sigma$ sets to $F_\sigma$ sets. I'm not sure how to demonstrate this. Specifically I'm not sure what ...