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

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3
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1answer
30 views

How can I prove that an ultrafilter induces a finitely additive measure?

If $\mathcal{U}$ is an ultrafilter on a set $X$, it can be defined a function $\mu_{\mathcal{U}}\colon\mathcal{P}(X)\to\{0,1\}$ such that, for all $A\subseteq X$ it holds $\mu_{\mathcal{U}}(A)=1$ iff $...
0
votes
1answer
20 views

Cancelling measures on both sides

Let $(X,\mathscr{A},\mu)$ be a measure space. Suppose I have $\mu(A)-\mu(B)\ge\mu(A)$ with $\mu(A)=+\infty$. Am I still allowed to cancel the $\mu(A)$'s from both sides and conclude that $\mu(B)=0$?
1
vote
1answer
46 views

Real Analysis, Folland Theorem 2.41/Exercise 53 The $n$ dimensional Lebesgue integral

Theorem 2.41/Exercise 53 - If $f\in L^1(m)$ and $\epsilon > 0$, there is a simple function $\phi = \sum_{1}^{n}a_j\chi_{R_j}$ where each $R_j$ is a product of intervals, such that $\int |f - \phi| &...
1
vote
1answer
44 views

The $n$-dimensional Lebesgue Integral Properties

Background Information: In the theorem here we fix a complete Lebesgue-Stiltjes measure $\mu$ on $\mathbb{R}$ associated to the increasing right continuous function $F$, and we denote by $M_{\mu}$ ...
3
votes
1answer
50 views

Problem 5, Chapter 3 from Stein and Shakarchi's “Real Analysis” on a version of the FTC

The problem reads: Suppose that $F$ is continuous on $[a,b]$, $F'(x)$ exists for every $x\in(a,b)$, and $F'(x)$ is integrable. Then $F$ is absolutely continuous and $$F(b)-F(a)=\int_a^b F'(x)\,...
0
votes
2answers
78 views

Equivalent random variables and sigma algebras

Consider two random variables $X$ and $Y$ defined on the same probability space $(\Omega,\sigma,P)$. We know that they are equivalent in the sense that $P(\{X \ne Y\})=0$. Let $A_X$ and $A_Y$ be the ...
2
votes
0answers
30 views

Why $\lim_{x\to \infty }Mf(x)=0$ where $Mf$ is the Hardy-Littlewood maximal function.

In a course it is written that since $Mf(x)$ decay to zero at infinity, the measure or $\left\{x\in\mathbb R^n\mid\left|Mf(x)\right|>\lambda\right\}$ is finite. I was looking for such a result on ...
0
votes
1answer
60 views

Measurable non-negative function is infinite linear combination of $\chi$-functions

Let $(X,\mathfrak{M},\mu)$ is measure space. Suppose that $f\geqslant 0$ is measurable function on $X$. We know that exists the increasing sequence of measurable simple functions $\{s_n\}$ such that $...
0
votes
1answer
67 views

Drawing numbers on a plane *uniformly*

I am not being very precise here, because I do not know what would be the more precise terminology. I would definitely appreciate any comments on my method and my terminology. By choosing uniformly ...
1
vote
0answers
18 views

When is the product of integrable functions integrable over the product space?

In Stein & Shakarchi's Fourier Analysis, p. 46, a change in order of integration in $\int\int f(y)g(x-y)e^{-inx}dydx$ for $f$ and $g$ Riemann-integrable is justified by adding the ad-hoc ...
0
votes
0answers
38 views

Do $\limsup A_n$ and $\liminf A_n$ always have probability 1 or 0?

Is it right that, for a sequence of events $(A_n)_{n}$, $\limsup A_n$ and $\liminf A_n$ have probability $1$ or $0$?\ My idea for $\limsup$ is for example the following: $$P(\limsup A_n) = \mathbb{E}(...
0
votes
0answers
23 views

*-homomorphisms on $L^{\infty}(X,\mu)$ implemented by measurable maps

Let $(X,\mu)$ and $(Y,\nu)$ be two probability measure spaces. To any measurable map $T:(X,\mu)\to (Y,\nu)$ with $\mu=\nu\circ T$, the following injective $*$-homomorphism is induced: $$\rho_T:L^{\...
0
votes
1answer
40 views

Total variation defined as Integral

Let $\phi: [a,b]\to \mathbb{R}$ a function absolutely continuous with derivative continuous. Show that $TV(\phi, [a,b])=T_a^{b}(\phi)=\displaystyle\int_a^{b} \left|\phi'(x)\right|dx.$ Hello, my ...
3
votes
1answer
33 views

Is the supremum of an almost surely continuous stochastic process measurable?

Let's take a stochastic process $(X_t)_{0\leq t \leq 1}$ and assume that the sample paths are almost surely continuous. Let us define $S \equiv \sup_{t \in [0,1]} X_t$. How can we show that $S$ is ...
1
vote
0answers
31 views

Invariant measure of Poincare map

Let $M$ be a smooth manifold and let $v$ be a tangent vector field on $M$. Consider a system of ordinary differential equations $$ \dot x = v(x), $$ in local coordinates $x = (x_1, \ldots, x_n)$. ...
2
votes
2answers
73 views

Lusin's theorem from Rudin RCA

Hi! Let me ask questions on Lusin's theorem from Rudin's RCA. $1)$ As we know $s_n=\varphi_n \circ f$ (from Theorem 1.17) then $$2^nt_n(x)=2^n(s_n(x)-s_{n-1}(x))=2^n(\varphi_n \circ f(x)-\varphi_{n-1}...
0
votes
1answer
41 views

Separable metric space with no isolated points.

Let $(X,d)$ be a separable metric space with no isolated points and $(X,\mathcal{B}(X),\mu )$ is a measure space such that $\mu(X)<\infty$. How to prove that for all $\epsilon >0$, there exists ...
17
votes
4answers
815 views

Most functions are measurable

My professor once said that if you did not use the axiom of choice to build a function $f : \mathbb{R}^n \to \mathbb{R}^m$, then it is Lebesgue measurable. To what extent this is true?
2
votes
1answer
47 views

Use Poincare Recurrence to show existence of $n$

Suppose $A\subset \mathbb N$ such that $d(A)=\lim_{n\to\infty}\dfrac{|A\cap [1,n]|}{n}>0$. Then show there exists $n\in\mathbb N$ such that $\overline{d}(A\cap (A-n))>0$ where $\overline{d}(B)=\...
4
votes
1answer
31 views

Theorem 2.22 from RCA Rudin

I read this interesting result from Rudin's book but I would like to clarify some confusing moments. As I understood $(\mathbb{R}^1, +)$ is group and $(\mathbb{Q}, +)$ is subgroup. He considers ...
1
vote
1answer
54 views

Real Analysis, Folland Problem 2.4.42, counting measure with convergence in measure

Problem 2.4.42 - Let $\mu$ be counting measure on $\mathbb{N}$. Then $f_n\rightarrow f$ in measure if and only if $f_n\rightarrow f$ uniformly. Attempted proof - Suppose that $\mu$ is a counting ...
0
votes
1answer
27 views

Can anyone explain an example related to vague convergence?

Let {$X_n$} be a sequence of random variable and {$\mu_n$} be a sequence of measures induced by {$X_n$} such that $\mu_n(B) = \mathbf{P}(X_n^{-1}(B))$. Suppose $X_n$ following a uniform distribution ...
1
vote
2answers
49 views

Real Analysis, Folland Problem 2.4.33 Modes of Convergence

Background Information: Theorem 2.30 - Suppose that $\{f_n\}$ is Cauchy in measure. Then there is a measurable function $f$ such that $f_n\rightarrow f$ in measure, and there is a subsequence $\{...
0
votes
1answer
33 views

A graph of any function has content zero?

I know that if $f$ is integrable them his graph have content zero, but if we consider $f : \mathbb{R} \rightarrow \mathbb{R}$ as $f(x) = 1$ if $x \in \mathbb{Q}$ and $f(x) = 0$ if $x \in \mathbb{R} - \...
2
votes
1answer
44 views

Integrals over subset of measure space

Let $(X, \mathcal{M}, \mu)$ be a measure space. Suppose $E \in \mathcal{M}$ and $f \in L^+$ where $L^+$ is a space of measurable functions from $X$ to $[0, \infty]$. $\int_E f$ is defined by $\int_X f\...
0
votes
1answer
21 views

Strongly measurable implies Borel measurable in separable space

Let $(M,\mu)$ be a measure space, $X$ be a Banach space, $f: M \to X$ be a function. $f$ is said to be strongly measurable if there is a sequence of simple functions $\{f_n\}\to f$ pointwisely a.e.. $...
1
vote
2answers
30 views

Does the collection $\{ \ [n, \infty) \ \mid \ n \in \Bbb N\}$ generate the Borel $\sigma$-algebra on $\Bbb R$?

Does the collection $\mathfrak C = \{ \ [n, \infty) \ \mid \ n \in \Bbb N\}$ generate the Borel $\sigma$-algebra on $\Bbb R$? I came across this problem in a paper. It seems like the answer is No. ...
1
vote
0answers
69 views

How to prove $f_n \in L^1$

I was trying to build a scheme to solve this kind of question: Let $D$ be a domain of $\Bbb R^n$ and $f_n\colon D \to \Bbb R$. Say if $f_n \in L^1(D)$. First of all I need to check that both $f_n$ ...
5
votes
2answers
57 views

Is $\bigcup_{x \in A} [x - 1, x + 1]$ Lebesgue measurable, where $A$ is a Lebesgue measurable subset of $\mathbb{R}$?

Suppose $A$ is a Lebesgue measurable subset of $\mathbb{R}$ and $$B = \bigcup_{x \in A} [x - 1, x + 1].$$Is $B$ Lebesgue measurable?
3
votes
1answer
35 views

Does there exist $G$ open and $F$ closed such that $F \subset A \subset G$ and $m(G - F) < \epsilon$?

Let $m$ be Lebesgue measure and $A$ a Lebesgue measurable subset of $\mathbb{R}$ with $m(A) < \infty$. Let $\epsilon > 0$. Does there exist $G$ open and $F$ closed such that $F \subset A \subset ...
3
votes
1answer
22 views

Measure on Borel $\sigma$-algebra of $\mathbb{R}$ is Lebesgue-Stieltjes measure

Let $\mu$ be a measure on the Borel $\sigma$-algebra of $\mathbb{R}$ such that $\mu(K) < \infty$ whenever $K$ is compact, define $\alpha(x) = \mu((0, x])$ if $x \ge 0$ and $\alpha(x) = -\mu((x, 0])$...
3
votes
2answers
48 views

Real Analysis, Folland 2.33 Egoroff's Theorem Modes of Convergence

2.33 Egoroff's Theorem - Suppose that $\mu(X) < \infty$, and $f_1,f_2,\ldots$ and $f$ are measurable complex-valued functions on $X$ such that $f_n\rightarrow f$ a.e. Then for every $\epsilon > ...
3
votes
2answers
52 views

Preimage of open set is Lebesgue measurable only if the function itself is measurable

It is a simple result in my book saying the proof is trivial, but I can not seem to show it. If someone can provide a hint just to help me begin my proof, it would be of assistance. Assume you know ...
1
vote
0answers
36 views

Convergence of measures in total variation sense

Suppose we have measures that defined over the set $\{0,1,2,..,C\}$. Let $\{\mathbb{P}_{n,m}\}$ be a sequence of measures. Suppose that for fixed $n$, $\mathbb{P}_{n,m}$ converges to $\mathbb{P}_n$ ...
2
votes
2answers
29 views

$\{A \subset X: \chi_A \in \mathcal{F}\}$ is a sigma algebra

Suppose $\mathcal{F}$ is a collection of real-valued functions on $X$ such that the constant functions are in $\mathcal{F}$ and $f + g$, $fg$, and $cf$ are in $\mathcal{F}$ whenever $f$, $g \in \...
1
vote
0answers
42 views

Is this $\sigma$-algebra necessarily uncountable or not? [closed]

Suppose $\mathcal{A}$ is a $\sigma$-algebra with the property that whenever $A \in \mathcal{A}$ is nonempty, there exist $B$, $C \in \mathcal{A}$ with $B \cap C = \emptyset$, $B \cup C = A$, and ...
4
votes
0answers
35 views

Is $\mathcal{B} = \{f^{-1}(A) : A \in \mathcal{A}\}$ a $\sigma$-algebra of subsets of $X$ or not? [closed]

Let $(Y, \mathcal{A})$ be a measurable space and let $f$ map $X$ into $Y$, but do not assume that $f$ is one-to-one. Define$$\mathcal{B} = \{f^{-1}(A) : A \in \mathcal{A}\}.$$Is $\mathcal{B}$ a $\...
2
votes
1answer
19 views

Ascending chain of monotone classes, $A$ necessarily in $\mathcal{M}$

Suppose $\mathcal{M}_1 \subset \mathcal{M}_2 \subset \ldots$ are monotone classes. Let $\mathcal{M} = \bigcup_{n = 1}^\infty \mathcal{M}_n$. Suppose $A_j \uparrow A$ and each $A_j \in \mathcal{M}$. Is ...
2
votes
2answers
37 views

Real Analysis, Folland Proposition 2.30 Modes of Convergence

Proposition 2.30 - Suppose that $\{f_n\}$ is Cauchy in measure. Then there is a measurable function $f$ such that $f_n\rightarrow f$ in measure, and there is a subsequence $\{f_{n_j}\}$ that converges ...
2
votes
0answers
26 views

If $\mu(E)\geqslant 0$ is it true that $E\in \mathfrak{M}$?

Suppose $(X,\mathfrak{M},\mu)$ be a mesure space. Let $E$ such that $\mu(E)\geqslant 0$. Can we conclude that $E\in \mathfrak{M}$? I think YES because $\mu$ is the set function with domain $\mathfrak{...
1
vote
2answers
23 views

Example of a set and two $\sigma$ algebras such that union is not a $\sigma$-algebra

What is an example of a set $X$ and two $\sigma$-algebras $\mathcal{A}_1$ and $\mathcal{A}_2$, each consisting of subsets of $X$, such that $\mathcal{A}_1 \cup \mathcal{A}_2$ is not a $\sigma$-algebra?...
5
votes
1answer
83 views
+50

Example of a set and monotone class where monotone class is not a $\sigma$-algebra?

What is an example of a set $X$ and a monotone class $\mathcal{M}$ consisting of subsets of $X$ such that $\emptyset \in \mathcal{M}$, $X \in \mathcal{M}$, but $\mathcal{M}$ is not a $\sigma$-algebra?
2
votes
0answers
47 views

Weak measurability of a set-valued map

Suppose that $A$ and $B$ are compact metric spaces. Let $f:A\times B\to B$ be a Borel measurable map (in the sense that for every Borel set $S\subseteq B$, $f^{-1}(S)$ belongs to the $\sigma$-algebra ...
1
vote
0answers
16 views

Is $\pi^1:C_c(G)\rightarrow \operatorname{End}(H)$ a homomorphism of the convolution algebra when $G$ is not unimodular?

Let $G$ be a Hausdorff locally compact group and $H$ a Banach space. Let $\pi:G\rightarrow \operatorname{GL}(H)$ be a representation and define $$\pi^1(\phi)v = \int_G\phi(x)\pi(x)vdx$$ for $v\in H$ ...
1
vote
0answers
58 views

Construction of Lebesgue measure in Rudin's RCA book

This theorem from Rudin's RCA book. Here's one moment from it's proof which seems to me very weird. Rudin states that equality $\lambda(E)=m(E)$ holds for all Borel sets. But I think that it's ...
0
votes
1answer
35 views

Lebesgue outer measure is countably subadditive but not finitely additive proof

I have read all the Qs on this but couldn't find a clear proof. How can I prove that Lebesgue's outer measure is not finitely additive? Thanks! Edit: I understand I must show that the measure of the ...
0
votes
0answers
95 views

convergence in distribution in Banach spaces

We let $\Omega$ be a compact metric space and consider $C(\Omega)$ to be the space of all continuous functions on $\Omega$. The dual space of $C(\Omega)$ can be seen as the set of all signed borel ...
2
votes
2answers
46 views

Real Analysis, Folland Proposition 2.29 Modes of Convergence

Background Information: $f_n\rightarrow f$ in $L^1$ $\Leftrightarrow$ $\forall\epsilon > 0,\exists N$ $\forall n\geq N$ $\int |f_n - f| < \epsilon$ A sequence $\{f_n\}$ of measurable complex-...
1
vote
1answer
59 views

If $X_n \stackrel{p, quickly}{\to} X$, then $X_n \to X$.

Probability with Martingales: Without using hint, can I just do something like this: http://math.stackexchange.com/a/1538503/140308 ? With using hint: By continuity of probability, I think ...
1
vote
0answers
16 views

The finite-dimensional distribution of a stochastic process

Let $K(s,t)$ be a real function over $T\times T$, where $T$ is arbitrary. $K$ has two properties: $K$ is symmetric ($K(s,t)=K(t,s)$). $K$ is nonnegative-definite ($\sum_{i,j=1}^k K(t_i,t_j)x_ix_j\...