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

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2answers
85 views

Union of sets as the union of disjoint sets - Does the proof $\forall n\in \mathbb{N}$ implies the proof for infinity?

I managed to prove that: $$\displaystyle\bigcup_{i=1}^n A_i=A_1\cup(A_1^c\cap A_2)\cup(A_1^c\cap A_2^c\cap A_3)\cup\dots\cup(A_1^c\cap\dots\cap A_{n-1}^c\cap A_n)$$ for $\forall n \in\mathbb{N}$. ...
2
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1answer
116 views

Relationship between measure theory and real analysis

Does measure theory generalize real analysis to abstract spaces? For example, you can now talk about convergence even on unordered fields.
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1answer
61 views

Assumptions involving product spaces

Suppose a random variable $X$ is distributed in $\mathbb{R}^{n}$ and we are given that $X' = (X_{1}', X_{2}')$ for $X_{i}$ distributed on $\mathbb{R}^{n_{i}}$. In general, what assumptions can I make ...
2
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1answer
71 views

$M\subset\mathbb{R}^n$ measurable. Show: There is a null set $N \subset \mathbb{R}^n$ and compact seq $K_m$ with $M=N\cup\bigcup_{m\in\mathbb{N}}K_m$.

Assignment: Let $M\subset\mathbb{R}^n$ be lebesgue-measurable. Show that, there is a null set $N \subset \mathbb{R}^n$ and a sequence $(K_m)_{m\in\mathbb{N}}$ of compact subsets $K_m \subset \...
3
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1answer
44 views

If $B\subset\mathbb{R}^n$ is measurable and $(x_l)_{l\in\mathbb{N}}$ is a bounded family, so $(B + x_l)$ is pairwise disjoint, then $\mu(B)=0$.

Assignment: Show that: If $B\subset\mathbb{R}^n$ is Lebesgue-measurable and if there is a bounded family $(x_l)_{l\in\mathbb{N}} \subset \mathbb{R}^n$ so that the family $(B + x_l)_{l\in\mathbb{N}}...
3
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1answer
60 views

Bounding $P(X \le \tau)$

I am trying to upper bounding $P(X \le \tau)$ where $X$ is non-negative r.v. and where $\tau \le 1$. I have become aware of the Reverse Markov inequality that says that, if $P(|X|\le a)=1$ then for $...
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1answer
66 views

$L_2$ as a Hilbert space and $\ell_2$

I know that, if measure $\mu$, with which measure space $X$ is endowed, has a countable base, i.e. if for any measurable $M\subset X$ there exists a measurable set $A_k\in\mathscr{A}$, where $\mathscr{...
2
votes
1answer
71 views

Help in a problem about Lebesgue integration inequality

Let $ (X,\mathcal{S},\mu)$ be a finite measure space, let $f$ be $\mathcal{S}$-measurable and let $E_{n}:= \{x\in X :n-1\le |f(x)|<n\}$ for $n=1,2,\dots$ Show that: $$f \in L_1\iff\sum_{n=1}^{\...
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2answers
77 views

Soft Question: Are sigma fields, fields?

I'm sorry if this is a foolish question but: Is a $\sigma$-field (of sets) a field (in the sense of algebra) if we only consider finite intersections and finite unions?
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1answer
140 views

every $\sigma$ algebra is a monotone class

I couldn't understand the monotone class theorem because of this lemma: "Every $\sigma$ algebra is a monotone class." How i can prove it?
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2answers
39 views

given $E_1, E_2, E_3, …$ prove that the measure of {$x \in X :$ x belongs to infinite number of sets $E_k$} is $0$

Say I have a $\sigma$-algebra $\mathcal{A}$ over a set $X$ and a measure $\mu$. Let $E_1, E_2, E_3, .... \in \mathcal{A}$ such that $\sum_{k=1}^\infty \mu(E_k)$ < $\infty$. let B = {$x \in X :$...
1
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1answer
45 views

If $f(\cdot,y)$ is measurable and $f(x,\cdot)$ is continuous, $\{x:|f(x,y)-f(x,0)| \leq \epsilon, \; \;\forall y <\delta\}$ is measurable

Suppose $\mu(X) < \infty$ and $f : X \times [0,1] \rightarrow \mathbb{C} $ is a function such that $f(\cdot,y)$ is measurable for each $y \in [0,1]$ and $f(x,\cdot)$ is continuous for each $x \in X$...
3
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1answer
190 views

Example of algebra that is not a $\sigma$-algebra

I understand that an algebra $F \subset 2^\Omega$ is called a $\sigma$-algebra if it additionaly satisfies: $(A_i)_{i \in \mathbb{N}}$ with $A_i \in F$ pairwise disjoint, then also $\cup_{i \in \...
1
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1answer
45 views

Use the Monotone Convergence Thm, to show $\displaystyle\int f \le \liminf \int f_n$

! (http://i.imgur.com/Zwt1m1n.png) I need to do the question at the top of this image. I figured out that $g_n$ is an increasing sequence that is pointwise convergent to $f$. i.e. I know $\lim g_n(x)...
1
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1answer
38 views

Where does the following intuition about $G_\delta$ sets fail?

Where does the following reasoning that $\mathbb{Q}$ is supposedly a $G_\delta$ set fail? "Proof": $\mathbb Q$ may be covered by selecting open sets $O_n$ such that $m(O_n)<\frac{1}{n}$ for all $...
1
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2answers
56 views

A question about inclusion of $L^r(\mu)$ spaces for different $r$ and different measures $\mu$

For some measures, the relation $r<s$ implies $L^r(\mu)\subseteq L^s(\mu)$ ; for others, the inclusion is reversed; and there are some for which $L^r(\mu)$ does not contain $L^s(\mu)$ if $r\ne s$. ...
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3answers
1k views

convolution of characteristic functions

Suppose $A$ and $B$ are measurable subsets of $\mathbb{R}$ of finite positive measure. Show that the convolution $\chi_A*\chi_B$ is continuous and not identically $0$. Use this to prove that $A+B$ ...
3
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0answers
71 views

“Uniform” Convergence in Distribution (bounded Lipschitz metric)

I have been thinking about the following problem. Let me know if the notation below makes sense. Let $\mathcal{P}$ denote the set of Borel probability measures on a metric space $(\mathbb{R}^{k}, ||\...
2
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1answer
69 views

Exercise on measure theory, (verification and suggestion)

Hi everyone I'd like to know if the following is correct and also I'd appreciate any suggestion to improve the argument. Thanks in advance For every positive integer $n$, let $f_n:{\bf{R}}\to [0,\...
2
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1answer
39 views

Limit of integral over set measurable

If $A\subset[0,2\pi]$ is measurable, prove that $$\lim_{n\to\infty}\int_A \cos (nx)\ dx=\lim_{n\to\infty}\int_A \sin(nx) \ dx=0$$ Please, any suggestions are welcome.
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0answers
94 views

Approximating simple summable function in measure space with countable base

Let $f:X\to \mathbb{Q}+i\mathbb{Q}\subset\mathbb{C}$, $f\in L_1(X,\mu)$ be a Lebesgue-summable function taking only finitely many values $y_1,\ldots,y_n\in \mathbb{Q}+i\mathbb{Q}$ on the sets $E_1,\...
4
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1answer
243 views

Product of Absolutely Continuous Measures is Absolutely Continuous

I am stuck on this problem from Folland's Real Analysis, Second Edition: For $j = 1, 2$, let $\mu_j, \nu_j$ be $\sigma$-finite measures on $(X_j, \mathcal{M}_j)$ such that $\nu_j <\!\!< \mu_j$. ...
1
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1answer
59 views

Showing Convergence

Let $(X,M,\mu)$ be a measurable space and $f$ be a real valued integrable function on $X$. Let $E_n=\{x\in X: f(x)\geq nq\}$ for every $n\in \mathbb{N}$ and fixed $q>0$ . Show that $\lim_{...
3
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1answer
62 views

(Hints please) Constructing a measurable set with the following property.

If $\delta >0$, $I_\delta=(-\delta,\delta)\in\mathbb{R}$, and $0\leq\alpha\leq\beta\leq1$, what hints do you have that would help me figure out how to construct a measurable set $E\subset\mathbb{R}$...
0
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1answer
28 views

Showing Convergence in measure with some condition. [closed]

Let $(X,M,m)$ be a finite measurable space and $\{f_n\}$ be a sequence of real valued measurable functions on $X$ . Let $$E_n=\{x\in X : f_n(x)\ne 0\}$$ for every $n\in \mathbb{N}$ . Show that if $ ...
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2answers
56 views

Integral Measures: Variation

Given a measure $\lambda\geq0$. Regard a real function $h:\Omega\to\mathbb{R}$ with $h\in\mathcal{L}$. Consider the real measure $\mu(E):=\int_E h\mathrm{d}\lambda$. Then its total variation ...
1
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2answers
40 views

Showing Convergence in $L^p$ norms

Let $X$ be a finite measure space and $1\le p<\infty$ and $\{f_n\}$ be a sequence in $L^p(X)$ such that coverge to $f$ in $L^p(X)$ . If there exists constant $K$ such that for every $n\in \mathbb{...
2
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0answers
47 views

Metric space with measure and a special property

Let $R$ be a metric space endowed with a (complete) measure $\mu$ satisfying the following condition: all the open and closed sets of $R$ are measurable and for any measurable set $M\subset R$ and any ...
0
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2answers
26 views

Moment generating function of bounded variables

According to the answer of this question a moment generating function exists if the random variable $X$ is bounded. The proof is not quite obvious to me. More formally, let $(\Omega,\mathcal{A},\mu)$ ...
2
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1answer
39 views

Analogue of Weierstrass Approsimation Theorem

The following theorems are well knows: Weierstrass Approximation Theorem Given a continuous function on $f\colon [a,b]\rightarrow \mathbb{R}$, there exists a sequence of real polynomials, which ...
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0answers
23 views

What is a function in $f\in L^n(\mathbb{R}^n)$ but $g(x)=\int_{|y|<1} \frac{|f(y)|}{|x-y|^{n-1}}dy$ is not in $L^\infty$

What is a function in $f\in L^n(\mathbb{R}^n)$ but $$g(x)=\int_{|y|<1} \frac{|f(y)|}{|x-y|^{n-1}}dy$$ is not in $L^\infty$. I have no idea where to start. Apparantly this is related to the ...
1
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2answers
121 views

Borel-Cantelli (proof and application)

Hi I was reading the second volume of the Tao's Analysis book and in one exercise he's asking for a proof of Borel-Cantelli If we have a sequence $s_n\in \Omega$ of measurable sets s.t. $\sum_{n\...
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2answers
194 views

Lebesgue integral $\int_{(0, \infty)} \frac{\sin x}{x} dm$ doesn't exist but improper Riemann integral exists

Show that the Lebesgue integral $\int_{(0, \infty)} \frac{\sin x}{x} dm$ doesn't exist but the improper Riemann integral $\int_{0}^\infty \frac{\sin x}{x} dx = \lim_{t\to\infty} \int_{0}^t \frac{\sin ...
4
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2answers
99 views

Folland Exercise 3.3: Stuck + Possibly missing $\sigma$-finite hypothesis?

This is Exercise 3.3 from Folland's Real Analysis, Second Edition stated exactly as it appears in the text: ''Let $\nu$ be a signed measure $(X, \mathcal{M})$. $(a)$ $L^1(\nu) = L^1(|\nu|)$. $(b)$ ...
1
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1answer
26 views

(Hints please) Prove that $\psi(E)=(\mu\times\lambda)(E)$ for every $E\in S\times T$.

Given that $(X,S,\mu)$ and $(Y,T,\lambda)$ are $\sigma$-finite measure spaces with the measure $\psi$ defined on $S\times T$ such that $\psi(A\times B)=\mu(A)\lambda(B)$ whenever $A\in S$ and $B\in T$....
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0answers
63 views

How to calculate expectation of Cantor distribution without using p = 0.5

The question goes like this: given F(x) is the distribution function of Cantor distribution, how to calculate $E(x) = \int xdF(x)$ without using the argument of the following: X is characterized by ...
0
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1answer
44 views

Non-differentiable in a null set

This is a problem from Stein's real analysis book that I have been working on. Show that exists a non-negative integrable f in $\mathbb{R}^{d}$ so that $\liminf_{m\left(B\right)\rightarrow0,x\in ...
2
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1answer
53 views

Is it sufficient to check only open intervals in order to prove that a real function is measurable?

Let $f : \mathbb R \to \mathbb R$. We say that $f$ is measurable if, for every $S \in \mathcal B$ where $\mathcal B$ is the Borel algebra on $\mathbb R$, we have that $f^{-1}[S] \in \mathcal B$. I ...
1
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1answer
46 views

Proove a fact about $\sigma$-algebras in preparation for Kolmogorov's zero–one law

Let $(J_n)_{n\in\mathbb{N}}$ be finite sets which monotonically increase to $I\cong\mathbb{N}$ and $(\mathcal{A}_n)_{n\in\mathbb{N}}$ be a family of $\sigma$-algebras. I want to show that it holds $$A:...
1
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1answer
68 views

Convergence of a sequence of functions integrated over a sequence of measures

I have real-valued functions $\{f_n\},f$ on a subset $X\subset \mathbb R^n$ that are equicontinuous and I have Borel measures $\{\mu_n\},\mu$. I have that For each fixed $m$, $\int f_m d\mu_n\to\int ...
1
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1answer
49 views

What do we mean when we say that a function $f$ takes the value $ \infty $?

What do we mean when we say that a function $f$ takes the value $ \infty $? In measure theory it is common to let mappings take values in the extended real number system. But still it doesn't make ...
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1answer
60 views

What are boundary effects in measure-theory?

I often read the term "boundary effects" which seem to be the reason to look at the interior $A^°$ and closure $\bar{A}$ of Borel subsets $A$ separately. What is so special about the boundary and what ...
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3answers
74 views

$f_n \to 0$ $ a.e.$ and $\lim \int f_n d\mu =0$ but $\sup_n f_n$ is not in $L^1$

Give an example of a finite measure space $(X,M,\mu)$ and a sequence of functions $f_n:X \to[0, \infty)$ such that $f_n \to 0$ $a.e.$ and $\lim \int f_n d\mu=0$ but $\sup_n f_n$ is not in $L^1$ I ...
1
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1answer
74 views

uncountable Lebesgue-null set of $[0,1]$ such that $1_N$ is not Riemann integrable

Give examples of the following and justify 1) uncountable Lebesgue-null set $N$ of $[0,1]$ such that $1_N$ is not Riemann integrable on $[0,1]$ 2) uncountable Lebesgue-null set $N$ of $[0,1]$ such ...
0
votes
1answer
102 views

Weak convergence on L^p

Let Let $X=[0,1]$ with the Lebesgue measure, find a sequence $\{f_n\}$ of measurable functions $f_n:X \rightarrow{ \mathbb{R} } $ such that: $f_n(x)\rightarrow{0}$ almost everywhere $x∈[0,1]$ $f_n$...
1
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2answers
66 views

Show that random walk is a random variable

I am working on this question. Suppose $\{X_n, n \ge 1\}$ are random variable on the probability space $(\Omega, \mathcal{B},P)$ and define the induced random walk by \begin{align*} S_0=0, \, ...
3
votes
1answer
81 views

Showing countable additivitiy of Lebesgue measure

The following is taken from the classic Probability and Measure by Patrick Billingsley, Theorem 2.2 (page 26 in the 3rd edition). I have a question on his proof, but I give the necessary defintions to ...
2
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1answer
214 views

Convergence in Total Variation Implies Convergence in Distribution

Suppose $X,Y$ are random variables. We define the total variation distance of random variables to be $d(X,Y)= \inf \{P(|X′−Y′|>0): X′,Y′$ are couplings of $ X,Y$ respectively$\}$. Does ...
1
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0answers
82 views

monotone convergence of $f_n$ plus weak convergence of $\mu_n$ implies convergence?

I would like to know if somebody is aware of some result that looks like the following. Let us consider the space $C_b(X)$ of continuous bounded function over a measurable space $X$. Suppose that: ...
5
votes
2answers
132 views

Why are some convergent Lebesgue integrals 'undefined'? [duplicate]

I sometimes read statements such as The integral $$\int_0^{\infty} dx \, \frac{\sin x}{x} $$ does not exist as a Lebesgue integral, because it is not absolutely convergent. But according to my ...