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

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Borel isomorphism between polish

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
15 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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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? ...
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1answer
21 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 ...
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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
28 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
29 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
18 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
26 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 ...
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16 views

I do not understand the hypothesis of the lebesgue decomposition theorem

I do not understand the hypothesis of the lebesgue decomposition theorem. Given a mesure in a sigma-algebra i do not understand why exists a set it is concentrate on.
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1answer
19 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 ...
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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
19 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 ...
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2answers
28 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 ...
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2answers
150 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
33 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 ...
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1answer
14 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 ...
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51 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 ...
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1answer
41 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. ...
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3answers
44 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~~~
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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 ...
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Convergence of average of translates of a function

Short version for people who don't like reading: Let $f\colon\mathbb{R}\to\mathbb{R}$ be $1$-periodic and $L^1$ on one period (or perhaps: measurable and bounded). Is it true that, for almost ...
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1answer
37 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 ...
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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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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? ...
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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
32 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., ...
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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
33 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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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 ...
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6 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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32 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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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. ...
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1answer
23 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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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 $(A_N)_{N \geq 0}$ ...
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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 ...
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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. ...
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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 ...
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27 views

Congruent measurable sets

I have a question regarding Congruent relations: In Euclidean geometry, two subsets of $\mathbb{R}^{d}$ are said to be congruent if one set can be mapped onto the other by translations and rotations. ...
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1answer
31 views

Weak convergence in $L^p$ equivalent to pointwise almost everywhere convergence

Can weak convergence of a sequence $f_n\in L^p(\Omega, \mu)$ to some $f\in L^p(\Omega, \mu)$ be characterised as almost everywhere pointwise convergence? Let us also assume the measure space is ...
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1answer
31 views

Equality in Conditional Jensen's Inequality

Conditonal Jensen's Inequality says that for a convex function $\varphi$, a random variable $X$, and a sub-sigma-field $\mathcal{F}$, $E[\varphi(X)\mid \mathcal{F}] \geq \varphi(E[X\mid ...
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1answer
29 views

Proving measurability in $\mathbb{R}^2$

I am given the problem: suppose for measurable, real-valued functions $f$ and $g$, and an open set $A \subset \mathbb R ^2$, prove that $\{x \in \mathbb R : (f(x),g(x)) \in A\}$ is a measurable set. ...
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25 views

$g(x) = sup_{α∈A} (f_α(x))$, $x ∈ E$ need not be a measurable function.

We know that if $(f_n)$ is a sequence of measurable functions on $E$, then $g = sup_n f_n$ defined as $g(x) = sup f_n(x)$, $x ∈ E_ n$ is a measurable function. Prove by an example that if $A$ is an ...
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1answer
42 views

Prove that there is no continuous function $f : \Bbb R → \Bbb R $ such that $f = χ_I$ almost everywhere on $\Bbb R$.

Let $I = [0,1]$ and $χ_I : \Bbb R → \Bbb R$ be the characteristic function on $I$. Prove that there is no continuous function $f : \Bbb R → \Bbb R $ such that $f = χ_I$ almost everywhere on $\Bbb R$. ...
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1answer
25 views

examples of random variables that the result of their preimage is not in F?

let's assume we have a probability space $(\Omega , F , P)$. and we have a random variable $X$ defined as : $X : \Omega \rightarrow \Bbb{R}$ and we also use a Borel set ($\mathcal{B}$).(making the ...
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1answer
21 views

Continuous map from $L^r(\Omega)$ to $L^s(\Omega)$.

The following theorem appears in the appendix of P.H. Rabinowitz monograph on Critical Point Theory: Let $\Omega \subset \mathbb R^n$ be bounded. Let $g$ be such that (i) $g \in C(\overline{\Omega} ...
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26 views

Exercise in “Elements of Integration” by Bartle

I found the problem below in Bartle's book "The Elements of Integration and Lebesgue Measure". I have not been able to solve it. All ideas are welcome. If $\phi$ is not uniformly continuous, then ...