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

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0
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2answers
32 views

Fatou's lemma. Examples with limit inferior $\neq$ lim.

I have problems with understanding Fatou's Lemma. What is the reason for using $\liminf$? Can someone please give an example where $\liminf \neq \lim$. When the reason does not depend on one of the ...
3
votes
0answers
18 views

Equality of measure sets of dynamical system

This is a homework question I have been crunching my brains on for a lot of time, but unfortunately I'm stuck. I would greatly appreciate any help! The problem is as follows: We have some continuous ...
0
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1answer
28 views

Generalised Holder ineq

Prove the following generalisation of Holder's inequality $$\int | u_1 \cdot ... \cdot u_N | d\mu \leq \|u_1\|_{p_1} \cdot ... \cdot \|u_N\|_{p_N}$$ for all $p_j \in (1,\infty)$ such that ...
1
vote
1answer
28 views

A question about independence of sigma algebras (generated by random variables)

Let $X_1, X_2, \ldots$ i.i.d random variables. Is it possible that $$\{X_{n+1} \in B\} \in \sigma({X_1, \ldots, X_n})$$ for some $B$? Why yes/not? I want to show that $\sigma(X_{n+1})$ and ...
0
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2answers
41 views

The cardinal of the set of all measures on $\mathbb{R}$

It is a very simple question that I don't know how to do: Let $M = \{\mu \colon \mathcal{B}(\mathbb{R})\to \mathbb{R} \colon \mu \text{ is a measure}\}$ $$|M| = \ ?$$ Any help will be appreciated.
1
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1answer
13 views

Comparing marginal on product space with other measure

My previous post Unifying the treatment of discrete and continuous random variable, got successfully answered and allowed me to get further in my results. However I am facing a question that I can't ...
2
votes
2answers
30 views

Family bounded in $\mathcal{L}^1$ has limit a.e.

Let $(X, \mathcal{F} , \mu )$ be a measure space. Suppose $\lbrace X_n \rbrace$ is a family of functions in $\mathcal{L}^1$, bounded in $\mathcal{L}^1$ i.e. there exist $K \geq 0 $ such that ...
3
votes
1answer
56 views

Strong law of large numbers for square-integrable and uncorrelated random variables with bounded variance

Let $(\Omega,\mathcal{A},P)$ be a probability space and $(X_n)_{n\in\mathbb{N}}$ be a sequence of square-integrable and uncorrelated (maybe we actually need independence) random variables $\Omega\to ...
0
votes
1answer
22 views

why the set $A=\{(x,y)\in\mathbb{R}\times\mathbb{R}:x-y\in E\}$ is $\mathcal{B}\times\mathcal{B}$-measurable

If $E\in\mathcal{B}$ , then the set $A=\{(x,y)\in\mathbb{R}\times\mathbb{R}:x-y\in E\}$ is $\mathcal{B}\times\mathcal{B}$-measurable, where $\mathcal{B}$ is the family of Borel subsets of $\mathbb{R}$ ...
0
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1answer
30 views

Does $a^2P(|X |\ge a )\le EX^2 $ hold when $a<0 $?

That is, does Chebyshev's inequality hold for when $a $ is negative? I have seen some authors to require that $a $ be positive, but when Reading the proof by Rick Durrett, I cannot see that this is ...
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0answers
28 views

Show that there is $f\in L^1(X,\mu)$ with $P(f)<\infty$ and $P(f_n-f)\to 0$ as $n\to\infty$ [on hold]

Could you please help me solving this old prelim problem. Any hints are appreciated
3
votes
1answer
27 views

Boolean algebra with measures

Let $A,B$ be two Boolean algebra with measures $m,p$ thereon, respectively such that the measure algebra $(A,m)$ is isomorphic to the measure algebra $(B,p)$. Suppose that we have two isomorphic ...
8
votes
2answers
87 views

Topology and Measures

I apologize if this question is a bit vague; I'm just wondering if there is a concept like what I'm talking about, or if I'm just lost. I'll start with just some thoughts. I looked a bit, and I don't ...
7
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2answers
37 views

Is every measurable set a measure-independent limit of open sets

My main question is Q1. Let $B$ be a Borel-measurable subset of $\mathbb R$. Is there a sequence of open sets $U_n$ independent of any measure such that for all Borel probability measures ...
0
votes
1answer
50 views

If a sequence $(f_n)$ converges in $L^2$, then $g'(x)\int_0^x f_n(t)\,dt$ converges in $L^1$

The first: Suppose $g$ is increasing and differentiable on $[0,1]$. For every $f\in L^2(0,1)$ define $f^*(x)$, for $x\in [0,1]$, by: $$f^*(x)=g'(x)\int_0^x f(t)\,dt .$$ If $f_n\to f$ in $L^2(0,1)$, ...
1
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1answer
39 views

Measurability of adapted processes

Let $(\Omega, \mathscr{A}, P)$ be a probability space, $(E, \mathscr{E})$ a measurable space and $X_t : \Omega \to E$, $t \geq 0$ a measurable stochastic process, i.e. the map $X : [0, \infty) \times ...
0
votes
0answers
19 views

For $E \subset \mathbb{R}$ and $\epsilon >0$, $\exists$ $(a,b)$ s.t. $\theta(E \cap (a,b)) \geq (1-\epsilon)|b-a|$ ($\theta$ Lesbegue Outer Measure)

In my notes this statement is left unproven. I want to show that for any measurable set $E \subset \mathbb{R}$ with $\theta(E)>0$, there exists an interval $(a,b)$ that covers $E$ arbitrarily ...
5
votes
1answer
81 views

Density of the rationals in the reals

While studying measure theory I have encountered the following set, $$U_\varepsilon=\bigcup_{n\in \mathbb{N}}(q_n-\varepsilon /2^n,q_n+\varepsilon/2^n),$$ where $(q_n)_{n\in \mathbb{N}}$ is an ...
0
votes
1answer
36 views

Definition of $\sigma$-algebra. Axioms.

""Def. A family $\mathcal F$ of subsets of $\Omega$ is said to be a $\sigma$-algebra on $\Omega$ if: (A.1) $\Omega\in\mathcal F$ (A.2) $\ A\in\mathcal F\implies\ A^c\in\mathcal F$ (A.3) $\ ...
-2
votes
0answers
23 views

a totally bounded subset of $\mathbb{L}_1(\mathbb{R}^{d})$ and Kolmogorov-Riesz compactness theorem

Let $\lambda$ be the Lebesgue measure on $\mathbb{R}^{d}$ , $\mathcal{F}$ a set of all probability densities $f$ such that $\mathcal{F}$ is a totally bounded subset of $\mathbb{L}_1(\mathbb{R}^{d})$ ...
1
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1answer
39 views

What does it mean to say the smallest σ-algebra?

I am just starting out on measure theory. What does it mean to say the smallest σ-algebra?
2
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2answers
46 views

A question about sum of n random variables

Let $X_1, \ldots, X_n$ be random variables. We know that $X_1, \ldots, X_n$ are $\sigma(X_1, \ldots, X_n)$ - measurable. But how about $X_1 + \cdots + X_n$? Is it $\sigma(X_1, \ldots, X_n)$ - ...
1
vote
2answers
33 views

If $f_n$ converges to $f $ in $p$-norm, then $f_n$ converges to $f$ in measure.

I want to prove that if $f_n$ converges to $f $ in $p$-norm, then $f_n$ converges to $f$ in measure. This is the proof: Suppose not. Then there exist $\epsilon>0,\delta> 0$ such that $μ \{x: ...
2
votes
2answers
53 views

What is a non-decreasing sequence of sets?

What is a non-decreasing sequence of sets and how come it can have a limit? It appear in a probability theory book
0
votes
1answer
38 views

Show that $g=\sum_{n=1}^{\infty } |f _{n+1 }-f _n | $ has $||g ||_p\le 1 $ if $||f _{n+1 }-f _n ||_p <2 ^{-n } $

Minkowskis inequality implies that $g _k=\sum_{n=1}^{k} |f _{n+1 }-f _n | $ has norm less than $1 $, and there is a hint to use Fatou's lemma to $g _k ^p$. Then $\int \lim \inf g _k ^p \le \lim \inf ...
0
votes
1answer
29 views

Showing convergence of a series almost everywhere

If $\sum_{k=1}^\infty a_k$ is convergent series of positive terms and $(\alpha_k)_{k\in \Bbb N}$ is a sequence of real numbers, then the series $$\sum_{k=1}^\infty\frac{a_k}{\sqrt{|x-\alpha_k|}}$$ ...
0
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2answers
31 views

Show $\sup_{y>0}\left|\int_0^\infty \int_t^\infty f(x,y) \cos\left(\dfrac{t}{y}\right)dx\,\,dt\right|<\infty$

Suppose $f$ is Lebesgue measurable on $[0,\infty)\times [0,\infty)$ and $g\in L^1([0,\infty))$. If $|xf(x,y)|\leq g(x)$ for all $y\in [0,\infty)$ prove that $$\sup_{y>0}\left|\int_0^\infty ...
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0answers
35 views

How do I see that $|f_n -f|^p = \lim_{n \rightarrow \infty} \inf |f_n - f_{n_k}|$? [on hold]

Let $(X, \mathcal E, \mu)$ be a measure space, $p \in [1, \infty]$ and $f, f_n \in \mathcal M(\mathcal E)$ Suppose $(f_n)$ is Cauchy in $\mu$-p-mean and $f_{n_k} \rightarrow f$ converge ...
0
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0answers
21 views

Problem with the definition of semi-ring and $\sigma$-sets

I have a problem with a statement I found concerning the definition of semi-ring and that of $\sigma$-set. So, here there is. Assume the definition of a semi-ring $\mathcal{S}$ over a non-empty set ...
1
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0answers
17 views

How to calculate convolution of function defining a measure

Given the function $F(t)=2-2e^{-t}$ defining a measure on $(\mathbb{R}_+,\mathfrak{B}(\mathbb{R}_+))$ and I want to calculate the convolution of this function with itself. I tried to do that by using ...
1
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0answers
16 views

Probability measure of rank-$r$ matrices

I have a question about the distribution of matrices with a specific rank. Consider $\mathcal{M}^{n\times m}$ the set of all $n \times m$ matrices with entries in some field $\mathbb{K}$. If I define ...
0
votes
2answers
41 views

Minkowski's Inequality in $L^\infty$ space

How can one show the inequality that $\|f+g\|_\infty ≤ \|f\|_\infty + \|g\|_\infty$? Can we use same real number $a$ for both $f$ and $g$ ? i.e, $$\|f\|_\infty = \text{ess} ...
1
vote
1answer
19 views

Cinlar Ex. 1.15: Trace space of a measurable space.

In constructing the trace space on a subset of a measurable space, it seems one has to assume that the subset is an element of the original measure space's sigma algebra, i.e., measurable in the ...
1
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2answers
69 views

Continuous functions of minimal norm

Let $C$ denote the set of continuos functions on $[0,1]$ with the supremum norm. $M\subset C$ such that $$\displaystyle\int_{0}^{1/2}f(t)\, dt-\int_{1/2}^{1}f(t)\, dt=1,\; \forall f\in M$$ Show ...
1
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1answer
17 views

Locally compact metric space, Urysohn, approximation

Let $E$ be a locally compact separable metric space, $\mathcal{B}(E)$ be the $\sigma$-algebra of $E$ and $m$ be a $\sigma$-finite borel measure on $(E,\mathcal{B}(E))$. Assumtion There exists a ...
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0answers
36 views

Hahn-Banach proof of existence of Haar measure

I'm reading these notes of Terry Tao on the Haar measure (and related topics) on a locally compact Hausdorff group $G$. When he goes through the construction of the Haar measure, he does so by way of ...
2
votes
2answers
38 views

prove that $F$ is $\mu\times\mathcal{L}$ measurable where $F(n,x)=\frac{(2n+1)^2\sin((2n+1)x)}{(n(n+1))^2}$

Let $\mu$ be the counting measure on $\mathbb{N}$ and $\mathcal{L}$ be the Lebesgue measure on $[0,\pi]$. Define the function $F$ on $\mathbb{N}\times\mathcal{L}$ by ...
6
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3answers
39 views

Borel measurable functions $f:[0,1]\to[0,\infty)$ which cannot be expressed as pointwise limit of nondecreasing sequence of step functions

An interval in this problem may be open, closed or half open. We regard a singleton $\{a\}$ as being an interval also. A step function is a real valued function on $\mathbb{R}$ which is a linear ...
0
votes
2answers
25 views

Looking for proof of theorem on complex measurable functions

In University I have been given the following result: If $f:X\to\mathbb{C}$ is a measurable function in $L^1(X,\mathcal{E},\mu)$ with $\mu$ being finite, and there exists a closed set ...
1
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2answers
58 views

Prove that a intergral over $\mathbb R$ is finite

Let $K\in \mathcal L_1(\mathbb R)$ and $f$ be measurable and bounded on $\mathbb R$ such that $\lim_{|x|\to \infty} f(x)=0$. Define $$F(x):= \int _{\mathbb R} K(x-s)f(s)\;ds \qquad (x\in \mathbb R)$$ ...
-1
votes
1answer
27 views

Dimension of rectifiable curve

Suppose $\Gamma$ is a rectifiable curve (means a curve with finite length), I want to prove that the Hausdorff measure of the intersection of it with closed subset $A\subset \mathbb{R}$ is 0, i.e ...
1
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0answers
53 views

Every projection of the square of the middle thirds Cantor set contains an interval

Let $C_\lambda$ the cantor set which is defined by the IFS $\{\lambda x,\lambda x+(1-\lambda)\}$ and also let $E=C_\lambda\times C_\lambda$.Suppose $\lambda =\frac 1 3$, we get the standard ...
1
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0answers
35 views

Approximation of Conditional Expectation with Respect to “Y” Using Simple Approximation of “Y”

Background. (TL:DR you can skip to Question. below.) This is a followup question to one of my previous questions (linked here) on this website. In short, the other question was about how to express ...
2
votes
1answer
19 views

$L_1$ convergence of $\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$

Does the sequence $f_n=\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$ on $(0,1)$ converge in $L_1$? It converges to zero pointwise and I think it converges in $L_1$ as well since ...
0
votes
1answer
30 views

The smallest filtration for which a sequence of random variables is adapted

Let $X_1, ..., X_n$ be a sequence of random variables. Show that $\hspace{60pt}$ $\mathcal{F}_n$ = $\sigma(X_1, ..., X_n)$ is the smallest filtration such that the sequence $X_1, ..., X_n$ is ...
1
vote
1answer
16 views

A relation between the inner and outer jordan measures

I'm studying measure theory and I was thinking about the following question: Is it true that whenever $A\subset B\subset \mathbb{R}^n$ are bounded, $$m^*(B-A)=m^*(B)-m_*(A)?$$ I have easily ...
0
votes
1answer
17 views

Infinite products of scaled indicator variables: almost sure convergence vs. uniform convergence of the sample mean

Let $\frac{X_i}{2}\sim Ber(0.5) \implies E[X_i]=1$, and let $Y_n=\prod\limits_{i=1}^n X_i$. Since the $X_i$ are iid, $E[Y_n]=1,\;\forall n<\infty$. However, something weird appears to be happening ...
0
votes
0answers
13 views

Automophism of G and Haar measure

Let $G$ be a locally compact group (written additively), $\lambda$ an automophism of $G$, and $\alpha$ a Haar measure in $G$. As the Haar measure is unique up to factor constant, $\lambda$ transform ...
1
vote
1answer
35 views

Measurable set limit

If $\forall n \in ℕ$ , $ f_n: (X,M) \rightarrow (\overline{\mathbb{R}},B) $ are measurable. (where X is any space, M is a sigma algebra, B is Borel sigma algebra) Prove that the set $A = \{x\in X: ...
2
votes
1answer
26 views

How to show that the function $g(x)=x|\sin(x^{-1/2})|$ is absolutely continuous?

I am having trouble showing the on $[0,1]$, $g(x):=x\mid\sin(x^{-1/2})\ \mid$ when $x>0$ and $0$ is $x=0$ is absolutely continuous. I was instructed to try: $\ m(A) < \delta \Rightarrow ...