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

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3
votes
5answers
2k views

Example of two functions that are equal almost everywhere?

We shall say that two functions $f$ and $g$ defined on a set $E$ are equal almost everywhere, and write $f(x)=g(x)$ a.e $x\in E$, if the set $\{x\in E: f(x)\neq g(x)\}$ has measure zero. I just ...
6
votes
1answer
495 views

Total variation of complex measure is finite

Let $\mu$ be a complex measure on a measurable space $(X, \Sigma)$. Let $|\mu|$ be the total variation of $\mu$, defined by $|\mu|(E) = \sup \left\{ \sum_{j=1}^{\infty} |\mu(E_j)| : ...
1
vote
1answer
105 views

Continuity of a function $F(t)= \int_{\Bbb R} f(x) \cos(tx) \, dx$

One of the past comp question Set $F(t)= \int_{\Bbb R} f(x) \cos(tx) \, dx$ , where $f \in L^1(\Bbb R)$ To prove $F(t)$ is continuous for any $t \in \Bbb R$ and $\lim_{t\rightarrow\infty} F(t) =0$ ...
2
votes
1answer
133 views

$f$ Borel measurable and and $f=g$ a.e (Lebesgue) but $g$ is not Borel measurable

I am reading Swartz's book (Measure, Integration and function spaces) and I come across an example 12, p. 73 which kind of make sense to me but not really. This deals with the mapping in the ...
9
votes
1answer
351 views

Convergence in $L^1$ space

Suppose that $f_{n}$ is a sequence of measurable functions, in a finite measure space, $f_{n}\to f $ in $m$-measure and that there exists $g$ in $L^1$ such that $\vert f_n\vert \le g$. Prove that ...
2
votes
3answers
883 views

Must $f$ be measurable if each $f^{-1}(c)$ is?

Suppose $f$ is a real-valued function on $\mathbb R$ such that $f^{−1}(c)$ is measurable for each number $c$. Is $f$ necessarily measurable?
1
vote
0answers
84 views

Image of a nonmeasurable set.

I have a set $S$ that is a nonmeasurable subset of $X=\{0,1\}^{\mathbb{N}}$ (with respect to the normed product measure on $X$. Now let $g:X\to[0,1]$ be defined by ...
2
votes
0answers
448 views

How to show a compact sets has finite measure?

This is a step used in proving Riesz representation theorem. However I cannot follow his short proof. For every compact set $K$, he construct an $f\in C_{c}(X)$ such that $0\le f\le 1$, and ...
2
votes
1answer
228 views

Proving $f$ is Absolutely Continuous

Let $\{f_{n}\}_{n}$ be a sequence of absolutely continuous function defined on $[0,1]$ such that $f_{n}(0)=0$ for all $n$. Assume that the sequence of derivatives $\{f_{n}^{`}\}_{n}$ is Cauchy in ...
2
votes
2answers
63 views

Convergence in $L^{p}$ spaces

Set $$f_n= n1_{[0,1/n]}$$ For $0<p\le\infty $ , one has that $\{f_n\}_n$ is in $L^p(\mathbb R)$. For which values of $p$ is $\{f_n\}_n$ a Cauchy sequence in $L^p$? Justify your answer. This was ...
3
votes
1answer
160 views

Proving $\phi$ is essentially bounded

This is one of the past qual question. Suppose $\phi$ is a real valued measurable function on $\mathbb{R}$ such that, for any $f$ in $L^{1} (\mathbb{R})$, the product $f\phi$ is also in $L^{1} ...
1
vote
1answer
83 views

$f$ converge in measure but fails to converge in $L^{1}$

I am thinking I can use the function $$f_{n}= n1_{[0,1/n]}$$ This will work beacuse $f_{n}\rightarrow 0 $ in measure because $$1/n \rightarrow 0$$ as $n$ gets bigger, but when we take the integral of ...
7
votes
0answers
228 views

Convergence of indicator functions in $L^2[0,1]$ when $m(\limsup(E_n)\setminus \liminf(E_n)) = 0$

I am trying to solve a qualifying exam problem. I would like to know what can be said about the convergence of the indicator functions $I_{E_k}$ in $L^2[0, 1]$ when it's known that $m(\limsup ...
4
votes
1answer
451 views

Lebesgue vs. Counting Measure on $[0,1]$

If $X = [0,1]$ and we let $m$ be the Lebesgue measure on $[0,1]$ and $\nu$ be the counting measure on $[0,1]$, then are the following statements true? The only subset $S \subset [0,1]$ that is of ...
0
votes
1answer
46 views

Does a series of functions that are 0 a.e. converge uniformly to the function 0 a.e. as well?

If $f_n = 0$ a.e. for all $n \in \mathbb{N}$, does $\sum f_n \rightarrow 0$ uniformly a.e. to 0 as well?
2
votes
1answer
107 views

Products of infinitely many measure spaces. Applications?

What are some typical applications of the theory for measures on infinite product spaces? Are there any applications that you think are particularly interesting - that make the study of this ...
2
votes
0answers
69 views

Application of LDCT to a Borel measureable function

Let $f:R\rightarrow R$ be a Borel measurable and assume $f$ is in $L_1$ [$0, T]$. Let $F(t)=\int$ $f(s) ds$. Show $d/dt$ $F(t)$ exists Lebesgue a.e on $[0, T]$ and is Borel measurable. I am reading a ...
3
votes
2answers
332 views

Calculating an Integral with respect to a Product Measure on $[0,1] \times [0,1]$

Let $S = [0,1] \times [0,1]$ and let $D = \{(x,y) \in S : x = y \} \subset S$. Let $\mu \times \nu = \lambda$ be the product of the Lebesgue measure $\mu$ on $[0,1]$ and the counting measure $\nu$ on ...
5
votes
2answers
2k views

Convergence in measure and almost everywhere

In a finite measure space, let $\{f_{n}\}$ be a sequence of measurable functions. Show that $f_{n} \rightarrow f$ in measure if and only if every subsequence $\{f_{n_{k}}\}$ contains a subsequence ...
2
votes
2answers
157 views

Prove an inequality in real analysis

Let $(X,F,u)$ be a measure space and suppose $\{ f_{n} \}$is a sequence of measurable functions with the property that for all $ n>1 $, $$ u \{ x\in X:|f_{n}(x)|\geq \lambda \} \leq ...
1
vote
1answer
438 views

Set of measure zero and limsup of its covering

Show that a subset $E$ of $R$ has measure zero iff there exists a sequence of intervals $\{I_{n}\}$ such that $a)\sum\ m(I_{n})<\infty; $ and $b)$ $x $ in $E$ implies $x$ lies in infinitely many ...
2
votes
1answer
125 views

Try to Understand Calculus Theorems, Real Analysis

I attempted some easy multiple-choice and T/F questinos to test if I am entirely clear about the topic before I do any proofy works. It's essential to be clear on these basic concepts and ideas. Let ...
8
votes
1answer
953 views

Existence of a Strictly Increasing, Continuous Function whose Derivative is 0 a.e. on $\mathbb{R}$

This proof is almost done except for the step of showing that the function's derivative is $0$ a.e. Let $I = \{[p_n, q_n]\}$ denote the set of all closed intervals in $\mathbb{R}$ with rational ...
4
votes
0answers
318 views

An exercise about the regular Borel measures

I want to prove following (from Folland, Ex. 3.26): If $\lambda$ and $\mu$ are positive, mutually singular Borel measures on $R^n$ and $\lambda + \mu$ is regular, then so are $\lambda$ and $\mu$. For ...
4
votes
1answer
696 views

Showing the Measure of the Limsup of a Countable Sequence of Measurable Sets is Zero

Let $(X, \mathcal{B}, \mu)$ be a finite measure space and suppose $\{A_n\} \subseteq \mathcal{B}$ s.t. $\sum_{n=1}^\infty \mu(A_n) < \infty$. Furthermore let $\underset{n \rightarrow ...
1
vote
1answer
148 views

Demonstrating Continuity Properties of $f(x) = \sum_{n=1}^\infty [a_n H(x - x_n)]$

Let $H: \mathbb{R} \rightarrow \mathbb{R}$ s.t. $H(x) = 0$ whenever $x < 0$ and $H(x) = 1$ whenever $x \ge 0$. Let $\sum_{n=1}^\infty a_n$ be a series of positive terms which converges. Let ...
1
vote
1answer
200 views

Is the arbitrary union if non-empty set measurable Lebesgue?

I'm working on the Vitali Covering Lemma. I'd like to see a dimostration of the statement in the title. I'm looking for a dimostration about the fact that the arbitrary union of set ( with non-epmty ...
2
votes
1answer
249 views

Lebesgue Measure vs. Product Measure

I am going through exercise problems of Rudin, and I came across this question. Find an example of a positive continuous function $f$ in the open unit square in $\mathbb{R}^2$, whose integral ...
5
votes
1answer
191 views

Showing that $\nu \ll \mu$ implies $\forall \epsilon > 0$, $\exists \delta > 0$ s.t. $\mu(A) < \delta \implies \nu(A) < \epsilon$

I am stuck on what I think may be the very last line of the proof I am seeking. Let $(X, \mathcal{B})$ be a measurable space which has associated with it the finite measures $\mu$ and $\nu$ s.t. $\nu ...
5
votes
1answer
81 views

Average Value for an uncountable set.

Let $S$ be an uncountable set of indexed real numbers. So the same number can occur more than once in the set (although with a different index). I don't assume that there is any ordering on $S$. Is ...
15
votes
1answer
580 views

Kakutani skyscraper is infinite

Karl E. Petersen's book "Ergodic Theory", chapter 2, exercise 9, on page 56 Prove that for any ergodic measure preserving transformation $T:X\rightarrow X$ on a non-atomic probability space $(X, ...
1
vote
0answers
282 views

Showing that the Countable Sum of Measures is Mutually Singular to another Measure

Is the following proof valid? Let $(X, \mathcal{B})$ be a measurable space and suppose $\lambda_n$ is a sequence of finite measures s.t. $\lambda = \sum_{n=1}^\infty \lambda_n$ is finite. I want to ...
4
votes
1answer
96 views

Is this function measurable on $\mathbb{R}$. General methods?

I'm trying to show the function $f:\mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{1}{[x (\log(x))^2]}$ for $x\in (0,\frac{1}{2})$ and $0$ otherwise is measurable. Are there any ...
6
votes
2answers
370 views

How to apply Borel-Cantelli Lemma?

Assume that we are given a sequence of continuous functions $f_n(x)$ on $[0,1]$. How to show the existence of a sequence $a_n$ and a set $A$ with $\mu(A^c)=0$ so that $$ \lim_{ n\to \infty} ...
5
votes
1answer
128 views

A measure theory question

Let $x_0,x_1,\ldots$ be an infinite sequence of real numbers with $x_n \in [0,1],\,\forall n\in\mathbb{N}$. Let $\mathcal{F}_0,\mathcal{F}_1,\ldots,$ be an infinite sequence of disjoint Lebesgue ...
5
votes
1answer
233 views

Rephrasing a Convergence Result to make use of the Borel-Cantelli Lemma

Let $X_n$ be a sequence of non-negative iid random variables. Is it true that the condition, $$\limsup_{n\rightarrow\infty} \frac{X_n}{n} = \infty \text{ almost surely}$$ is equivalent to the ...
1
vote
1answer
345 views

The indicator function of an open set is an increasing limit of a sequence of continuous functions

Let $G$ be an open set on some measure space $(\Omega,\Sigma, \mu)$. Show that the indicator function of $G$ is an increasing limit of a sequence of continuous functions. I understand that this is a ...
3
votes
2answers
179 views

Why for a compact metric probability space, any Borel subset can be approximated by compact set?

Let $X$ be a compact metric space with a Probability Borel measure $\mu$. Let $C$ be any Borel subset of $X$. Then for any small positive number $a$, we can find compact set $K$ such that $K$ is ...
2
votes
1answer
163 views

Is the limsup of a sequence of measures also a measure?

Given a sequence $(\mu_{n})_{n\in\mathbb{N}}$ of $\sigma$-finite measures on the measurable space $(Ω,\Sigma)$, is the $\limsup_{n}\mu_{n}$ also a $\sigma$-finite measure? Clearly, ...
1
vote
0answers
110 views

Product measure: why are those the two 'extreme' cases?

Lets say we start off with two measure spaces $(X, \mathcal{A}, \mu), (Y, \mathcal{B}, \nu)$ and suposse want to form the product measure. It can happen that there is more than one 'product' ...
0
votes
1answer
241 views

$G_\delta$ bounded set measurable

I've seen a similar question on this forum before, yet i could not find a solution to this particular problem. Could you give me some tips on how to proceed with the solution? Here's the statement of ...
3
votes
1answer
139 views

Proving that $\mu(A)=\mu(B)$ if $A\cap E=B\cap E$ and $\mu(E)=1$ where the measure of the space is $1$

I have the following exercise I wish to solve: Let $(X,S,\mu)$ be a measure space s.t $\mu(X)=1$. Let $\mu^{*}$be defined on $X$ by: $\forall E\subseteq ...
1
vote
1answer
180 views

Lebesgue Integral of ${x^2}$ over $[0,1]$

I know its just simple using Riemann Integral. I do not know how can I show the integral of ${x^2}$ over $[0,1]$ equals to $\frac13$ using definition of Lebesgue integral.
1
vote
1answer
154 views

Conditional Expectation and identically independent distributed random variables.

I have a problem that can be resolved if i show that $$E(\varepsilon_k\mid\sigma(\varepsilon_1,\ldots,\varepsilon_{k-1}))=E(\varepsilon_k)$$ where $\varepsilon_1,\ldots,\varepsilon_k$ $\sim ...
3
votes
1answer
315 views

Proving that $(\mu^{*})^{*}=\mu^*$ for a measure $\mu$

I am not sure I am using the standard definitions so I will open by defining what I need: Let $X$ be a set, $\nu:\, \mathscr{P}(X)\to[0,\infty]$ will be called an external measure if ...
1
vote
1answer
39 views

How to formulate an integral on the probability space of a Markov shift?

For a problem I am currently working on, I would like to calculate the integral over the space $(\{1,2,\dots,n\}^\mathbb{N},\mathcal{F},\nu)$ where $\nu$ is the markov measure. My problem is that ...
0
votes
2answers
576 views

Using the Memoryless Property to Explain the Expected Value of the Maximum of iid Exponential RVs

Let $T$ and $V$ be independent random variables that are exponentially distributed with rates $\lambda$ and $\mu$. Consider their maximum, $$W = \max(T,V)$$ From the answer to a previous post, I ...
5
votes
1answer
273 views

Equi-integrability of a single function: is it the same as summability?

Let $(\Omega, \mathcal{M}, \mu)$ be a measure space and let $f\ge 0$ be a measurable function on $\Omega$. Suppose that $f$ satisfies the following properties: For all $\varepsilon > 0$ there ...
2
votes
0answers
187 views

Inner Approximation of Lebesgue Measurable Set

Can every bounded Lebesgue measurable set be approximated from the inside by countably many disjoint closed rectangles? A citation or proof would be nice.
2
votes
2answers
54 views

Proving that if $\nu(A)=0$ then $A$ is $\nu$- measurable

I am not sure I am using the standard definitions so I will open by defining what I need: Let $X$ be a set, $\nu:\, \mathscr{P}(X)\to[0,\infty]$ will be called an external measure if ...