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

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2answers
25 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
votes
1answer
57 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
27 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
12 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
votes
0answers
43 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
votes
1answer
36 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
42 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
24 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 ...
3
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0answers
22 views

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 ...
1
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1answer
35 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
votes
1answer
20 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: $$\left\|\sum_{j=1}^Jf_j\right\|^p_p\leq C\sum_{j=1}^J\|f_j\|_p^p$$ ...
0
votes
2answers
26 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
votes
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
28 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
25 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
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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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 ...
0
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1answer
24 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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0answers
27 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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0answers
5 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 ...
0
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1answer
28 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
21 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
votes
2answers
37 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
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1answer
19 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 ...
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1answer
26 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. ...
1
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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 ...
0
votes
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. ...
0
votes
1answer
24 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 ...
1
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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 $\Bbb R$. ...
0
votes
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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0answers
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 ...
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1answer
27 views

Weak convergence of measures and compact sets

Suppose that we have a sequence of probability measures $\{ \mathbb{P}_n \}$ converging weakly to a probability measure $\mathbb{P}$. Suppose that $M$ is a metric space with a compact subset $K$. I ...
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0answers
30 views

Does the quadratic covariation process define a measure?

In the context of stochastic integration (when we define the space $L^2(M)$), we define the (possibly infinite) measure $$P_M = P \otimes [M]$$ by $$E_M[Y] = E\left[\int_0^\infty Y_s(\omega) ...
2
votes
1answer
48 views

Weak convergence of probability measures and uniform convergence of functions

I am stuck on Problem 4.12 of Karatzas and Shreve's book Stochastic Calculus and Brownian Motion: Suppose that $\{ \mathbb{P}_n \}$ is a sequence of probability measures on $(C[0, \infty), ...
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1answer
35 views

$\mu(E)\ge \nu(E)\ \forall E\in A\ \Rightarrow\ \mu(\cup E)\ge\nu(\cup E)$? Here $\mu,\nu$ are probability measures on a $\sigma$-algebra.

$\mu(E)\ge \nu(E)\ \forall E\in A\ \Rightarrow\ \mu(\cup E)\ge\nu(\cup E)$? A. Here $\mu,\nu$ are probability measures on a $\sigma$-algebra $M$ on a set $X$. We can assume that arbitrary unions of ...
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1answer
53 views

Show $ (\int_{-\infty}^\infty \sqrt{p}\sqrt{q}d\mu)^2\leq 2 \int_{-\infty}^\infty \min\{p,q\}d\mu $

Consider a random variable $X$ in $(\Omega, \mathcal{F}, \mathbb{P})$. Let $p,q$ be two densities with respect to a measure $\mu$ in $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ where ...
1
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1answer
12 views

Partition generated $\sigma$-algebra

I saw this example given as a $\sigma$-algebra in various places. It goes like this: Let $X$ be a set and assume that the collection $\{A_1,\dots, A_N\}$ is a partition of $X$. Then the collection ...
2
votes
2answers
57 views

Does the following condition implies full outer measure?

Let $X \subseteq 2^{\omega}$ be a set of positive Lebesgue measure. Suppose that for every $\eta, \nu \in 2^{<\omega}$ of the same length, the measure of $X$ above $\eta$ is the same as the measure ...
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0answers
66 views

Concavity of the $n$th root of the volume of $r$-neighborhoods of a set

Let $A$ be a closed subset of $\mathbb{R}^n$. For $r>0$, let $A_r$ be the $r$-neighborhood of $A$, namely the set $\{x:\operatorname{dist}(x,A)\le r\}$. Is the function $f(r) = \mu(A_r)^{1/n}$ ...
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votes
0answers
17 views

Lebesgue measure of region under curve

Let $(X,\Sigma,\mu)$ be a $\sigma$-finte measure space and $f \in L^+(X,\Sigma)$. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$. Theorem: Define the area under the graph of $f$ to be ...
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1answer
44 views

Suppose $X_n \to_{p} X$, if $\limsup_n E|X_n|^r \leq E|X|^r$, how can I show that $X_n \to_r X$?

If I have that $X_n \to_p X$ (convergence in probability), and if $\limsup_n E|X_n|^r \leq E|X|^r$ for all $r \geq 1$, how can I show that $X_n \to_r X$ (this means $L^{r}$ convergence)? My goal is to ...
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vote
1answer
34 views

If we showed that $\mu(F_n)<\infty$ for all $n\in \mathbb{N}$, can we get $\cup_{n \in \mathbb{N}}F_n<\infty$?

If we showed that $\mu(F_n)<\infty$ for all $n\in \mathbb{N}$, can we get $\cup_{n \in \mathbb{N}}F_n<\infty$? The problem is the following: In the solution of Folland chapter 1 exercise 14, ...
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0answers
23 views

why is the collection of all finite subsets of $\mathbb{R}$ not a $\sigma-ring$

It says the definition of a $\sigma-ring$ is if $A,B \in \mathcal R$ then $A \setminus B \in \mathcal R$ and if $ A_{n} \in \mathcal R \forall n \in \mathbb{N}$ then $\cup_{1}^{\infty}A_{n} \in ...
0
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0answers
21 views

Probability density function above a given value. $\{ f(x) > c\}$

Say $X$ is a stochastic variable with a distribution $\nu$ and $f$ is the corresponding Lebesgue-measurable density. If I want to calculate a set $$A = \{ x \in \mathbb{R} \ | \ f(x) > c \}$$ for ...