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

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18 views

Showing that $\mu$ is a measure when continuous from above

Statment Let $\mu$ be a set function defined on a $\sigma$ -algebra. Show that $\mu$ is a measure given that $\mu \geq 0$, $\mu(\emptyset)=0$, $\mu$ is continuous from above and countably additive. ...
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0answers
5 views

For a stopping time $T$, prove that $X^T_t = \mathbb{E}\left[X_T\mid \mathcal{F}_t\right]$

We have a sigma-algebra $\mathcal{F}=\mathcal{F}_{\infty}$, a stopping time $T$ and an integrable random variable $X$ and define a martingale by $X_t = \mathbb{E}[X \mid \mathcal{F}_t], ...
3
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1answer
16 views

$\sigma$-algebra generated by weak topology in Hilbert Space

In general, if we have $H$ Hilbert space, and equipped with the weak topology, say $\tau^\ast$, is $\sigma(\tau^*)=\mathcal{B}$?, where $\mathcal{B}$ is the usual Borel $\sigma$-algebra I suspect it ...
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1answer
10 views

Question about Folland's proof of extension-of-premeasures theorem

Here is an excerpt from Folland's Real Analysis. I don't understand why the calculation $\nu (E)\leq \sum _n \nu (A_n)=\sum _n \mu_0(A_n)$ implies $\nu(E)\leq \mu (E)$. Why is this? The $A_n$ are not ...
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0answers
7 views

Local Riesz Potential estimate in terms of Maximal Function

For $f \in L^1_{\text{loc}}(\mathbb R^n)$, and fixed $R > 0$ we defined the local Riesz potential by $$I(x) = \int_{B(x,R)} \frac{f(y)}{\lvert x-y \rvert^{n-1}} d\lambda (y), \hspace{1cm} x \in ...
0
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0answers
23 views

Categorically deducding measurability of sections

Two lemmas which are often proved in elementary measure theory courses are that sections of measurable sets are measurable, and sections of measurable functions are measurable. Note $E_x= \left\{y\in ...
2
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0answers
45 views

How can I show that one of $m(A)$ or $m(\Bbb{R}\setminus A)$ is zero?

Let $A \subseteq \Bbb{R}$ be Borel measurable, and $T$ a dense subset of $\Bbb{R}$. Suppose for every $t \in T$ that $$m((A+t)\setminus A)=0,$$ where $m$ is the Lebesgue measure. Then I want to show ...
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1answer
25 views

Borel Sigma Algebra generated by (a, b] [on hold]

Let {(a,b]} be a class of sets, where a and b is an element of R, a < b, a can be negative infinity and b can be positive infinity. Let B be the sigma algebra generated by the class. Show that the ...
1
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1answer
16 views

If two functions differ on a set of positive measure, must their essential infima differ, too?

Suppose $f,g : [0,1]^2 \to [0,1]$ are measurable functions differing on a set $P$ of positive Lebesgue measure. Claim: there exists $A, B \subseteq [0,1]$, each of positive measure, such that ...
4
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1answer
36 views

Prove that $\sigma(F)=\Omega$

Let $F=\{A_1,...,A_n\}\subset P(X)$; $F_a=A_1^{a_1}\cap A_2^{a_2}\cap\cdots \cap A_n^{a_n}$ $ a=(a_1,...,a_n)\in \{0,1\}^n$ $$A^{a_i} = \begin{cases} A, & \text{if } a_i=0 \\ A^c, & ...
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0answers
16 views

Borel isomorphism between polish spaces

In my lecture on stochastics the following result has been used: For any uncountable Polish space $X$ there is a Borel isomorphism between this space and the real line. I was not able to find a ...
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1answer
18 views

Proving a variation of DCT

As homework, I was given the following problem. Suppose $f_n\overset{\text{a.e}}{\rightarrow}f$, and for each $n$ there's a $g_n\in L^1$ satisfying $|f_k|\leq g_k$. Prove that if $g=\lim _n g_n$ is ...
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0answers
21 views

any sum of sets open\nullset is a set of the same form

I'm curious how can one prove that any sum of sets $G\setminus N$, where $G$ is open and the Lebesgue measure of $N$ is 0, is a set of the same form. it is easy for countable sums, but in general? ...
2
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1answer
25 views

Does there exists a $\sigma$-algebra $\mathcal{F}$ such that$f$ is $\mathcal{F}/\mathcal{B}$ measurable iif $f$ is continuous?

Let $f$ be a function from $(\mathbb{R}, \mathcal{F}) \rightarrow (\mathbb{R}, \mathcal{B})$, where $\mathcal{F}$ is a sigma-algebra and $\mathcal{B}$ denotes the Borel sigma-algebra. Does ...
2
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1answer
21 views

Are the limits of a.e. equal sequences of measurable functions equal a.e.?

I haven't seen the following fact in any textbook or reference, which either means that it is trivial, or that it's false. Hopefully it is the former. I've attempted a proof: Claim: Let $f_n, g_n : ...
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1answer
29 views

Proving $f_n\rightarrow f$ such that $\sup_n \| f_n \|_1 \leq K$ implies $\| f \|_1\leq K$

Looking back at my notes from class, I see: Claim. $f_n\rightarrow f$ such that $\sup_n \| f_n \|_1 \leq K$ implies $\| f \|_1\leq K$. It appears after the statement and proof of Fatou's lemma but I ...
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1answer
33 views

Continuity of $F(x)=\int_{(-\infty,x]}fd\lambda$

For a homework assignment I was told to prove that given $f\in L^1(\mathbb R)$, the following function is continuous $$F(x)=\int_{(-\infty,x]}fd\lambda.$$ I thought to use DCT and show sequential ...
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1answer
20 views

Example of a Lebesgue unmeasurable function f such that f*f is Lebesgue measurable

Giv an example of a Lebesgue unmeasurable function $f:[0,1]\rightarrow \mathbb{R}$ such that $f^2$ is Lebesgue measurable.
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1answer
31 views

Poincare' recurrence theorem in measure theory.

I want to propose a problem, it's a version of Poincare' Recurrence Theorem, it's very similar to another problem proposed in this forum, but a bit different: Another version of the Poincaré ...
2
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1answer
20 views

Non-Borel a.e limit of Borel functions

As a homework assignment I'm supposed to prove or disprove Borel measurability is closed under a.e convergence. I think this is not true because the Borel $\sigma$-field is not complete. However, I'm ...
2
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0answers
20 views

Measurable real functions from $\sigma$-algebra generated by finite partitions

I was given the following homework problem. Let $f:X\rightarrow \bar{\mathbb{R}}$ a set function and $X$ be a measurable space whose $\sigma$-algebra is generated by a finite partition $E_1,\dots ...
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1answer
22 views

uniform Distribution on uncountable Lebesgue $0$-Sets

I know that for every measurable Set A it is possible to create a uniform Distribution on A if - A is finite - A is not a lebesgue 0-set and its not possible for infinite countable sets so I ...
2
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2answers
32 views

Why is Monotone Convergence Theorem restricted to a nonnegative function sequence?

Monotone Convergence Theorem for general measure: Let $(X,\Sigma,\mu)$ be a measure space. Let $f_1, f_2, ...$ be a pointwise non-decreasing sequence of $[0, \infty]$-valued ...
5
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2answers
156 views

Are there sets of zero measure and full Hausdorff dimension?

I would like to ask the following: Are there "many" sets, say in the interval $[0,1]$, with zero Lebesgue measure but with Hausdorff dimension $1$? The motivation for this question is the ...
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2answers
34 views

If $F$ is finite then is $\sigma(F)$ also finite?

Let $F\subset X$ be a finite family of sets of $X$. Is the sigma-algebra generated by $F$ ($\sigma(F)$) also finite? I was trying to use induction: If $F$ has one element say $A$ then ...
0
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1answer
17 views

Representing $C(X)$ as multiplication operators on $L^p$

Suppose that $X$ is a compact Hausdorff space and I represent $C(X)$ isometrically in $B(L^p(X,\mu))$ as multiplication operators for some finite positive regular Borel measure $\mu$. If I remember ...
4
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0answers
55 views

Lebesgue versus Riemann integrable

Can a Lebesgue measurable function be modified on a set of first category so as become continuous except on a set of Lebesgue measure zero? OR Can a Baire-measurable function be modified on a set of ...
3
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1answer
48 views

Why a function in a measure space is random variable?

Let $(\Omega,\mathcal{F})$ be a measure space and $X$ mapping from $\Omega$ to $\mathbb{R}$. Assume that $X^{-1}((a,b])\in \mathcal{F}$ for all intervals. Prove that $X$ is a random variable. ...
3
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3answers
47 views

A footnote about outer measure

This is the theorem about in Royden's real analysis book. And in the book there is a footnote I am confusing: Can anyone help me understanding it with examples~~~
0
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1answer
26 views

Null set of reals

I'm having trouble to understand a step of a proof. Let $S$ be a subset of $\mathbb{R}$. Prove that $S$ is null (Lebesgue measure). The book says the following: "It is clear that we can restrict ...
6
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1answer
128 views

Convergence of Riemann sums of a periodic function

Short version for people who don't like reading: Let $f\colon\mathbb{R}\to\mathbb{R}$ be $1$-periodic, measurable and bounded. Is it true that, for almost all $x$, the average of $f(x)$, ...
1
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1answer
39 views

Weak convergence and $\lim_{n\to \infty} \|f_n\|_{L^p}=\|f\|_{L^p}$ imply norm convergence.

Consider a $\sigma$-finite measure space $(X,A,\mu)$ and $f,f_n\in L^p(\mu)$ with $1<p<\infty$. If $f_n \stackrel{w}{\to} f$ and $\lim_{n\to \infty} \|f_n\|_{L^p}=\|f\|_{L^p}$ hold, then ...
0
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1answer
22 views

Stochastic process independent of its future

Are there examples of predictable stochastic processes $X$ such that their past is independent of their future? More formally, such that $\sigma\{X_s | s\in (0,t]\}$ is independent of $\sigma\{X_s | ...
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0answers
15 views

Triangle inequailty for $L^p$ norm to power $p$

I would like to prove the sharp estimate for the $L_p$ norm to power $p$ with $1\leq p <\infty$. What is the constant $C$ here: $$\left\|\sum_{j=1}^Jf_j\right\|^p_p\leq C\sum_{j=1}^J\|f_j\|_p^p$$ ...
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2answers
27 views

Is it true in general that $\int_{|X| \leq \epsilon} |X|^r \, d\mathbb{P} \leq \epsilon^r$?

If I have that $X$ is a random variable, for $\epsilon > 0$, and $r \geq 1$, is it true that: $$\int_{|X| \leq \epsilon} |X|^r \, d\mathbb{P} \leq \epsilon^r.$$? If so, is there a reason why? ...
0
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1answer
16 views

Show that Uniform$(1,5)$ is neither singular nor absolutely continuous with respect to Uniform$(0,3)$.

Actually, I'm just studying singular continuity, absolute continuity.I know the definitions.And have solved few very basic sums. Now, in this problem, I'm not understanding what does this 'with ...
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2answers
34 views

Prove that if $\mu (A) = \nu(A)$ for all $A \in s$, then this also holds for all $A \in M(s)$

Let $s$ be a collection of subsets of $X$. Assume that $\mu$ and $\nu$ are two measures on $M(s)$. Prove that if $\mu(A) = \nu(A)$ for all $A \in s$, then this also holds for all $A \in M(s)$, i.e., ...
3
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0answers
31 views

Riesz-Type Representation Theorems for Convex Functionals

It is well known that any positive linear functional $L$ on the spase $C_c([a,b])$ of functions continuous on an interval $[a,b]$ with compact support can be written as \begin{align*} ...
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2answers
34 views

$f$ has a zero integral on every measurable set. Prove $f$ is zero almost everywhere

I am trying to solve the following exercise: Let $f$ be integrable. Assume that $\int_A f d\mu = 0$ for every measurable set $A$. Prove that $f = 0$ a.e. [$\mu$]. I have the following proof but it ...
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0answers
24 views

Inner measure of a set

My question is problem 15 of chapter 3 of Wheeden and Zygmund which states: If $E$ is measurable and $A$ is any subset of $E$, show that $m(E)=m_{*}(A) + m^{*}(E-A),$ where $m_{*}$ and $m^{*}$ denote ...
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1answer
27 views

Rudin's RCA, Chapter 2 Definitions

I am currently reading Rudin's RCA, and I have some questions about a particular definition he uses in chapter 2: The following passage is taken from Rudin's RCA, page 47, section 2.15: "A measure ...
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28 views

E is measurable, then measure of E is the sum of the inner measure of a subset of E and the outer measure of the complement of the subset in E

If E is a measurable and A is any subset of E, show that $|E|=|A|_i+|E-A|_e$ where |E| is the measure of of E, $|A|_i$ is the inner measure of A, and $|E-A|_e$ is the outer measure of $E-A$. I have ...
0
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1answer
8 views

total variation for closed set zero if measure is zero on closed subsets

Let $\mu$ be a complex borel measure on $\Omega$, $|\mu|$ its total variation and $A \subseteq \Omega$ a closed set s.t. for each closed set $B\subseteq A$ we have $\mu(A)=0$. Now does it hold that ...
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1answer
33 views

Show that $\sigma(\mathcal{H})$ is equal to $\mathcal{P}(\mathbb{N})$.

Let $\mathbb{N} = \{1,2,3,4,\dots \}$ and define the sets $A_k \subset \mathbb{N}$ by $$ A_k = \{k,2k,3k,\dots \} $$ for $k = 1,2,\dots$. We denote by $\mathcal{H}$ the collection $\{A_1, A_2, ...
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0answers
24 views

Definition of integrability for sequences

My text book does not provide much about counting measures and integration. So I decided to setup integration on space $(N , P(N) , \mu_c ,R)$ myself imitating the construction of Lebesgue integral. ...
1
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1answer
25 views

Indicator function and liminf and limsup

Can anyone please explain why the following is true? And what is the intuition behind it? $$\chi_A(x) = \begin{cases}1 &, x \in A\\ 0 &, x \notin A.\end{cases}$$ Then we have ...
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0answers
14 views

Convergence of stochastic processes via convergence of infinitesimal generators

Given a sequence of sequence processes $(X_N(\cdot))_{N \geq 0}$, I want to show this sequence converges to another process $X(\cdot)$ by considering that the sequence of generators $(A_N)_{N \geq 0}$ ...
2
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2answers
41 views

If I have that $\limsup_{n}E|X_n|^{r} \leq E|X|^{r}$, is that enough to show that $\{|X_n|^{r}:n\geq 1\}$ is uniformly integrable?

If I have that $\limsup_{n}E|X_n|^{r} \leq E|X|^{r}$, is that enough to show that $\{|X_n|^{r}:n\geq 1\}$ is uniformly integrable? I am not sure here if the limsup condition here is as strong as if I ...
1
vote
1answer
20 views

Counting measure on sigma algebra power set of natural numbers .

My text book does not provide much about counting measures and integration. So I decided to setup integration on space $(N , P(N) , \mu_c ,R)$ myself imitating the construction of Lebesgue integral. ...
2
votes
1answer
42 views

Lebesgue integral of vector-valued function?

In Bernt Øksendals stochastic differential equations he says that if we have a random variable $X:\Omega\rightarrow\mathbb{R}^d$. He defines the expectation: $E[X]=\int_\Omega ...