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

learn more… | top users | synonyms (1)

1
vote
1answer
58 views

Finite measure spaces with a total closed set

Endowing $R$ with a finite borel measure. How to find a closed set with its total measure and every closed subset of it has minor measure?
6
votes
1answer
265 views

A Haar measure via the Lebesgue measure on $\Bbb R^d$

$\newcommand{\d}{\mathrm{d}}$ This the Exercise 3, Chapter 11 of the Gerald B. Folland book Real Analysis: Let $G$ be a locally compact group that is homeomorphic to an open subset $U$ of $\Bbb ...
10
votes
4answers
3k views

Interpretation of limsup-liminf of sets

What is an intuitive interpretation of the 'events' $$\limsup A_n:=\bigcap_{n=0}^{\infty}\bigcup_{k=n}^{\infty}A_k$$ and $$\liminf A_n:=\bigcup_{n=0}^{\infty}\bigcap_{k=n}^{\infty}A_k$$ when $A_n$ are ...
2
votes
3answers
550 views

Advantage of accepting non-measurable sets

What would be the advantage of accepting non-measurable sets? I personally feel that non-measurable sets only exist because of infamous Banach-Tarski paradox...
2
votes
1answer
182 views

Convergence in the absence of DCT, uniform integrability, and $\limsup E(X_n)$

This question is extended from Resnick's exercise 5.13 in his book A Probability Path. Let the probability space be the Lebesgue interval: $(\Omega=[0,1],\mathcal{B}([0,1]),\lambda)$ and define ...
2
votes
1answer
91 views

Convergence in expectation for: $X_n=\sum\limits_{k=1}^n\frac{(-1)^k}{k^2}x_k$

Here is another self-study exercise that I am struggling mightily with: $X_n=\sum\limits_{k=1}^n\frac{(-1)^k}{k^2}x_k$ where $\omega=(x_1,x_2,...)$ is a series of Bernoulli (1/2) trials. I am told ...
2
votes
1answer
160 views

Existence and finiteness of Lebesgue integral for: $f(x)=x^{-1}(e^{-x}-e^{-1/x})$

I think I am getting a little better at these MCT, DCT-type exercises. The issue is to show/prove the existence and finiteness (if they apply) to the following function: ...
1
vote
1answer
152 views

Partition of a probability measure in a continuous and atomic part

Let $(\mathbb{R}, \mathcal{B}, \mathbb{P})$ be a probability space. I want to show that $\mathbb{P}$ can be written as $\mathbb{P} = \mu + \nu$, where $\mu$ is a continuous measure (no atoms) and ...
2
votes
2answers
235 views

$\sigma$-algebra induced by $\{\{1\},\{2\},\ldots,\{n\}\}$ and the limit $n \rightarrow \infty$

Let $\Omega = \mathbb{N}$ be the natural numbers and $\mathcal{E}_n = \{\{1\},\{2\},\ldots,\{n\}\} \subset \Omega$. $\mathcal{A}_n = \sigma (\mathcal{E}_n)$ shall be the $\sigma$-algebra induced by ...
3
votes
1answer
908 views

Limit inferior/superior of sequence of sets

Let $(\Omega, \mathcal{A}, \mu)$ be a measure space, where $\mu(\Omega)< \infty$. Further $(A_n)_{n \in \mathbb{N}}$ is a a sequence of $\mathcal{A}$-measurable sets. I want to prove, that $$ \mu ...
3
votes
1answer
778 views

lim sup of sequence of continuous function from $[0,1]\rightarrow [0,1]$

$f_n:[0,1]\to [0,1]$ be a continuous function and let $f:[0,1]\to [0,1]$ be defined by $$f(x)=\operatorname{lim\;sup}\limits_{n\rightarrow\infty}\; f_n(x)$$ Then $f$ is continuous and ...
3
votes
1answer
626 views

Egoroff's theorem in Royden Fitzpatrick (comparison with lemma 10)

Hi math stackexchangers, I have a question about the difference between two math statements (for reference they can be found in Royden Fitzpatrick pages 64-65). Egoroff's Theorem: Assume $E$ has ...
0
votes
1answer
131 views

Seeking clarification of Lebesgue definition given for $\int _{0}^{1}x^{-a}dx$

I came across the example "Show that $\int _{0}^{1}x^{-a}dx$ exists as a Lebesgue integral, and is equal to $1/(1-a)$, if $0 < a < 1$; but is infinite if $a\geq 1$. The Lebesgue definition of ...
3
votes
1answer
158 views

the integral of a periodic function

Consider $f:\mathbb{R}\to \mathbb{C}$ a bounded and $1$-periodic function, and $g \in L^1(R)$ then $$\lim_{n\to \infty} \int _{R}g(x)f(nx)dx=\int_0^1f(s)ds\int_R g(t)dt.$$ I think the fact that $f$ ...
4
votes
1answer
2k views

Weak convergence and weak star convergence.

If region $\Omega$ is bounded and $u_n$ has weak star convergence in $L^\infty ( \Omega)$ to some $u\in L^\infty(\Omega)$ , does it imply that $u_n$ converges weakly in any $L^p(\Omega) $ ? I think ...
3
votes
1answer
110 views

Evaluation of $L^p$ function

Functions in $L^p$ are only defined $µ$-almost everywhere, so for a given evaluation point $x$, $F(x)$, $f\in L^p$ can be changed to any value, so in general it would not be well-definied to just ...
2
votes
1answer
266 views

Another question about the generated $\sigma$-field

Suppose I have a probability space $(\Omega,\mathcal{F},P)$ and the set $$\mathcal{C}:=\{F=\sum_{i=0}^nf_i\mathbf1_{(t_i,t_{i+1}]}|n\in\mathbb{N},f_i:\Omega\to\mathbb{R} \mbox{ measurable and ...
1
vote
2answers
58 views

General nonatomic measure that cannot be expressed as an integral

I read in a paper (Kingman — Poisson Processes, 2005) that: In most cases the mean [of an inhomogenous Poisson process on a set $A$] is given in terms of the rate function $\lambda(x)$ on $S$ by ...
2
votes
1answer
60 views

$a\mapsto \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map

Proof for $a\in (0,1)$ that $a\to F(a)= \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map, if $f\in L^p(X)$ for all $p\geq1$ in some measure space $(X,\mu)$. I'd like to prove that for all ...
0
votes
2answers
146 views

Lebesgue integrable p and q

Be $E=[0,\infty]$ and $1\leq p\leq \infty$, Exhibit a function $f$ that $f\in \mathcal{L}^p$ and $f\notin \mathcal{L}^q$ when $q \neq p$ I try with simple functions but i dont know if are ...
1
vote
1answer
92 views

Question on whether a set is measurable

Let $A$ be a non-Lebesgue measurable set, and let $B=[0,1]\subseteq\mathbb{R}$. Show that $C=A\times B$ is non-measurable. I try use the regularity of the lebesgue measure, but don´t work, maybe ...
1
vote
0answers
141 views

Fixpoint of monotone operators

Let $X$ be some set and let $F$ be the set of all functions with a domain $X$ and a range $[0,1]$. We consider $F$ to be a partially ordered set with $f\leq g$ if and only if $f(x)\leq g(x)$ for all ...
2
votes
0answers
194 views

Jordan decomposition measure and its distribution function

Let $f: \mathbb{R} \rightarrow \mathbb{R}$, and for $x \in \mathbb{R}$, let $T_f(x)=\sup \left\{\sum_{j=1}^N \left|f(x_j)-f(x_{j-1})\right|\right\}$, where the supremum is taken over all $N \in ...
3
votes
0answers
122 views

Question about proof of completeness of $L^p$

In my notes we prove completeness of $L^p$ by showing that if $\sum\|f_k\|_p < \infty$ then $\sum f_k$ converges in $L^p$. (That's a lemma we prove a bit earlier, namely that $(V, \|\cdot\|)$ is ...
0
votes
1answer
142 views

generating $\sigma$-field of a set

Let $X=(X_t)$ be a stochastic process and we define the raw filtration by $F=(\mathcal{F}_t)$, where $\mathcal{F}_t:=\sigma (X_s;s\le t)$ Now I want to prove that $\sigma (\mathcal{C})=\mathcal{F}_t$, ...
2
votes
1answer
226 views

Hausdorff Measure- Lower semi-continuity

By definition , when we are given a set $A \in \mathbb{R}^n$ , $$ H_\delta^{n-1} (\partial A ) = \inf \left\{ \sum_{j=1}^{\infty} \alpha_{n-1}\frac{1}{2^{n-1}} [\operatorname{diam}(U_j)] ^{n-1} ...
4
votes
1answer
185 views

Probabilistic Example which might not be defined on a Polish space

Probabilist often work on Polish spaces. Does somebody know an ("non-exotic") example, for which it is not possible to work on a Polish space, but instead one has to work on a general measurable ...
1
vote
1answer
131 views

Lebesgue measure on $\mathbb{R}/\mathbb{Z}$

I was reading a (brief) introduction about measure theory today and came across the following statement: (Lebesgue measure on $\mathbb{R}/\mathbb{Z}$): There is a unique probability measure $\mu$ ...
2
votes
2answers
87 views

Question about $L^p$ spaces

Suppose $1<p<\infty$ and let $L^1$ and $L^p$ denote the usual Lebesgue spaces on $[0,1]$. Let $$A=\{f\in L^1:\|f\|_p\leq 1\}.$$ Show $A$ is closed in $L^1$. I took a sequence $\{f_n\}$ ...
2
votes
0answers
201 views

$p$-norm is a convex function of $p$

For what measures $\mu$ and what intervals $(a,b) \subset (1,\infty)$ is the function $$p\mapsto \|f\|_p =\left(\int |f|^p d\mu \right)^{2/p}$$ a convex function of $p$ on $(a,b)$ for all $f\in ...
4
votes
1answer
300 views

Weak*-convergence of regular measures

Let $K$ be a compact Hausdorff space. Denote by $ca_r(K)$ the set of all countably additive, signed Borel measures which are regular and of bounded variation. Let $(\mu_n)_{n\in\mathbb{N}}\subset ...
6
votes
1answer
539 views

stopped filtration = filtration generated by stopped process?

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky: Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
1
vote
2answers
75 views

Measure spaces such that the semi-norm is a norm

In my lecture notes there is the following exercise: "Characterize those measure spaces $(X, B, \mu)$ on which the semi-norm $\|f\| = \int_X |f| d \mu$ defined on $L^1(X) = \{ f \mid f \text{ ...
3
votes
1answer
102 views

How to show that $\int \phi \,d\mu=-\int\phi'(x)f(x)\,dx$

Assume that $f\colon \Bbb R \rightarrow\Bbb R$ is left-continuous nondecreasing and let $\mu$ be a Borel measure in $\Bbb R$ such that $\mu([a,b))=f(b)-f(a)$ for $a<b$, $a,b \in\Bbb R$. I would ...
5
votes
3answers
1k views

Could someone remind me why is incorrect to switch an infinite sum and an integral?

Could someone jog my memory on this? The order of operation between an $\int$ and $\sum_{n\in \mathbb{N}}$ is not always interchangable? Note that the sum is an INFINITE sum Why is it that $\int ...
8
votes
1answer
528 views

Inclusion of $L^p$ spaces

Let $X \subset L^1(\mathbb{R})$ a closed linear subspace satisfying \begin{align} X\subset \bigcup_{p>1} L^p(\mathbb{R})\end{align} Show that $X\subset L^{p_0}(\mathbb{R})$ for some ...
4
votes
1answer
374 views

Infinite-dimensional translation-invariant measure

Why is there no translation-invariant measure on an infinite-dimensional Euclidean space? Is there a reasonably short, insightful proof? I am interested in an infinite-dimensional space with a ...
0
votes
1answer
433 views

interchange sum and integral

suppose I have a family of i.i.d standard normal random variables $Y_{n,k}$ and I define $X^N_t:=\sum_{n=0}^N\sum_{k=1}^{2^n}Y_{n,k}\phi_{n,k}(t)$ for $t\in [0,1]$ where $\phi_{n,k}$ are the Schauder ...
1
vote
1answer
128 views

Explanation of statement needed (Bochner-style integral, Fubini's theorem, etc.)

I am reading a paper. They define $$ L_{p,q}(Q) = \{ u \in L_p((0,T); L_q(Y)) : u(t, \cdot) = 0 \text{ on } Y \backslash Y_t \text{ for a.e. $t \in (0,T)$}\}$$ with norm $$\lVert u ...
1
vote
0answers
111 views

Completeness of $L^1$

I have written a proof that $L^1$ is complete. Can you read it and tell me if it's right? Thanks. To show $L^1$ is complete we use the following fact: Fact: If $f_n$ is a sequence in $L^1$ such that ...
2
votes
1answer
49 views

Give a example about invariant ergodic measure and quasi-symmetric mapping

Is there a example $(X,f,\mu)$ such that $X$ is a closed subset of Euclidean space, $f$ be a quasi-symmetric mapping but not a Lipschitz mapping, $f(X)=X$, $\mu$ is a finite measure on $X$ that is ...
1
vote
2answers
1k views

set in $\mathbb{R}$ which is not a Borel-set [duplicate]

Possible Duplicate: Lebesgue measurable but not Borel measurable Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly if i start from the topology of $\mathbb{R}$, i.e. all ...
2
votes
1answer
97 views

When does non-negativity of the integral of a function imply that the function itself is non-negative?

Let $(\Omega,\Sigma)$ be a measurable space and $(\omega_k)_{k\in\mathbb{N}}$ a sequence of elements of $\Omega$. Let $$ \mathcal{M}:=\left\{\sum_{k=1}^\infty a_k\cdot\delta_{\omega_k}: ...
0
votes
1answer
79 views

Question about an implication in a theorem

There is the following theorem: If $(f_n)$ is a sequence in $L^1$ such that $\sum \|f_n\|_1 < \infty$ then (1) $\sum f_n $ converges almost everywhere (i.e. $\sum f_n(x) = K_x < \infty $) ...
2
votes
2answers
139 views

Wrong proof of convergence almost everywhere

Can you tell me where the mistake is? If $(f_n) \in L^1$ is a sequence of functions such that $\sum_n \|f_n\|_1 < \infty$ I can prove that $f(x) = \sum_{n=1}^\infty f_n(x) < \infty$ for all $x ...
1
vote
2answers
419 views

Sequence of Uniformly Bounded functions

Consider a sequence $\{ f_k \}_{k=1}^{\infty}$ of locally-bounded functions $f_k: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$. Assume the following. For any sequence $\{X_k\}_{k=1}^{\infty}$ of ...
4
votes
2answers
418 views

Measurable homomorphism of $\mathbb{T}$ into $\mathbb{C}^\times$

Working through Katznelson's An Introduction to Harmonic Analysis and have been stumped by the following problem for the past few days: Show that a measurable homomorphism of ...
2
votes
3answers
802 views

Give an example of a measure which is not complete

Give an example of a measure which is not complete ? A measure is complete if its domain contains the null sets.
2
votes
2answers
83 views

Prove $\mu(\{|f-\mu f|>K\}) \le \frac{1}{K^2}(\mu f^2 -(\mu f)^2)$

Let $f$ be a random variable on a probability space $(\Omega, \Sigma,\mu)$ where $\mu f^2 < \infty$. How would I prove (or disprove) that $$\mu(\{|f-\mu f|>K\}) \le \frac{1}{K^2}(\mu f^2 -(\mu ...
4
votes
2answers
162 views

Uniqueness of extension of zero measure

Let $(\Omega,\mathscr F)$ be a measurable space with two probability measures $\mu, \nu: \mathscr F\to[0,1]$ defined over it. Suppose that $\mathscr C\subset\mathscr F$ is some class of sets and $$ ...