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

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1answer
19 views

Absolute value of complex Radon measure

Let $X$ be a locally compact Hausdorff space. To simplify matters we assume $X$ has a countable base. Let $\mathcal B$ be the $\sigma$-algebra generated by the set of open subsets of $X$. A Borel ...
0
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1answer
56 views

First uncountable ordinal

I am a beginner of ordinals and I don't know any powerful techniques in it. I come across with a problem about the first uncountable ordinal like this. Let $X$ be a set of uncountable cardinality. ...
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3answers
298 views

What's the Lebesgue measure of this set?

The following citation is from Folland's Real Analysis. Let $m$ denote the Lebesgue measure on $\mathbb{R}$ and $\{ r_j \}$ be an enumeration of the rational numbers in $[0,1]$, and given $E > ...
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0answers
14 views

Optimal $A\in \Sigma$ that maximizes an objective

Let $([0,1],\Sigma, \lambda)$ be a probability space. For any given $B\in \Sigma$, $K\in [0,1]$ and $f\in L^2(\lambda)$ with $f(x)\in[0,1]$ for all $x $, $$\max_{A\in \Sigma}\int_A f(x) d\lambda(x)- ...
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1answer
24 views

Is the support of the Gaussian finite or infinite?

Considering that as $x \to \pm \infty$ ; $e^{-\frac{x^2}{2}} \to 0$, is the support finite or infinite? A simple enough question, but enough to make me scratch my head. I feel that it's almost a ...
1
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0answers
17 views

Locality of Borel measure

Let $X$ be a locally compact Hausdorff space. To simplify matters we assume $X$ has a countable base. We denote by $\mathcal B(X)$ the $\sigma$-algebra generated by the set of open subsets of $X$. A ...
1
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0answers
26 views

Empirical distribution generates exchangeable $\sigma$-algebra

I have a problem understanding the following statement from Klenke, p. 234: If we write $\Xi_n(\omega) := \xi_n \bigl(X(\omega)\bigr) = \frac{1}{n} \sum^n_{i=1} \delta_{X_i(\omega)}$ for the ...
2
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0answers
29 views

For any given value x, are there uncountably many (countably infinite) binary sequences (ones and zeroes) whose limiting relative frequency is x

I have the following question, and given few proofs (provided by friends, professors, and my myself) which seem to work, I suspect the answer is yes: But I am still not completely sure. The question ...
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0answers
36 views

$\sigma$-algebra of events invariant under permutations

Let $X = (X_n)_{n\in\mathbb{N}}$ be a stochastic process with values in $E$. For $n \in \mathbb{N}$, define $$\mathcal{E}'_n := \sigma\bigl(F : F : E^\mathbb{N} \rightarrow \mathbb{R} \text{ ...
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0answers
16 views

Subspace of $L^p(X,\Sigma,\lambda)$

Consider $R$-valued functions in $L^p(X,\Sigma,\lambda)$, where $X=X^1\times X^2$, $\Sigma=\Sigma^1\times \Sigma^2$ and $\lambda=\lambda^1\times \lambda^2$ For given $i$, does the subsapce $M=\{f\in ...
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0answers
51 views

Puzzles in a proof

From a previous link in MSE: Prove the set of which sin(nx) converges has Lebesgue measure zero (from Baby Rudin Chapter 11), the question states Suppose that $\{n_k\}$ is an increasing sequence ...
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0answers
18 views

condition of equal almost everywhere for functions

I'm thinking of this question because I have seen a lot of theorems in real analysis stated with the condition "almost everywhere". But I wonder how different two functions can be given they are equal ...
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2answers
95 views

Prove $f_n\to f$ on $[a,b]\implies \int_a^b|f_n-f|\to 0$

Suppose $f,f_n$ are measurable and uniformly bounded on $[a,b]$. Prove $f_n\to f$ on $[a,b]\implies \int_a^b|f_n-f|\to 0$ Attempt: We note that since $f$ and $f_n$ are bounded and are ...
4
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1answer
36 views

measure of a set invariant by rational translation

Say that a measurable subset $A$ of $[0,1]$ is ${\mathbb Q}$-stable if $a+q\in A$ whenever $a\in A,q\in{\mathbb Q}$ and $a+q\in [0,1]$. Obviously, $\emptyset$ is ${\mathbb Q}$-stable and has measure ...
1
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2answers
42 views

How can we use Fubini's theorem to simplify $\int_0^r\frac 1{\sigma^{n-1}}\int_0^\sigma\rho^{n-1}f(\rho)\;d\rho\;d\sigma$?

Let $f:[0,\infty)\to\mathbb{R}$ and $R>0$. How does Fubini's theorem imply $$\int_0^r\frac ...
2
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1answer
26 views

Prove operator is isometry

Let $(X,\mathcal{A},m,T)$ be a probability preserving transformation. Prove that the operator $U:f\mapsto f\circ T$ satisfies $$ \|Uf\|_{p}=\|f\|_{p} $$ for every $1\le p<\infty$. My idea: $$ ...
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2answers
34 views

Compute $\lim_{n \to \infty}\int_{0}^{\infty} \frac{n \sin(\frac{x}{n})}{x(1+x^2)}dx$ using Dominated Convergence Theorem (just require checking)

The question ask me to compute $$\lim_{n \to \infty}\int_{0}^{\infty} \frac{n \sin(\frac{x}{n})}{x(1+x^2)}dx$$ by using Dominated Convergence Theorem. Here is what I did. I know as $n \to ...
0
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0answers
12 views

How to show measures extend from $\mu_0$ isn't unique

Let $\mathscr A = \sigma(B)$ be the $\sigma$-algebra generated by $B$ where $$B = \{ (a,b] \cap \mathbb Q \mid -\infty \leq a \lt b \leq \infty \}$$ And $B$ is an algebra. Now define a function ...
0
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1answer
29 views

tail event of a symmetric random walk

$S_n$ is a symmetric random walk. How do I show that the event $A_M=[\limsup\limits_{n\to\infty}S_n\leqslant M]$ is in the tail sigma algebra $\cap_{m>n}\sigma(X_m,X_{m+1},...)$? I would be really ...
0
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0answers
17 views

Does every measurable function create a partition of its domain?

Let $(S,\mathcal{S})$ and $(T,\mathcal{T})$ be two measurable spaces. Does every measurable function $f\colon S \to T$ create a partition of $S$? If $T$ is finite $T=\{C_1,\dotsc,C_n\}$ then this is ...
0
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1answer
22 views

a measurable set intersect a compact set

A set $E\subset \mathbb{R}$ is measurable if given $\epsilon >0$ there is an open set $G$ and a closed set $F$ such that $F\subseteq E \subseteq G$ and $m(G-F)<\epsilon$, where $m$ is the outer ...
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0answers
20 views

Show that $\lim_{n \to \infty}\Phi_{n}(x)=f(x)=\lim_{n \to \infty}\Psi_n(x)$

Let $f: [a,b] \to \mathbb{R}$ be bounded and continuous for a.e $x(\lambda)$. (a) Let $\{P_n\}_{n \ge 1}$ be any sequence of partitions of $[a,b]$ such that $P_{n+1}$ is a refinement of $P_n$ and ...
0
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1answer
20 views

The convergence set of a sequence of functions can be expressed in terms of upper and lower envelopes

let $f_n:\mathbb R\to[0,\infty)$ be a sequence of functions. Its lower envelope sequences are defined as $\underline{f_n}(x)=\inf\{f_k(x):k\geq n\}$. And its upper envelope is defined similarly except ...
2
votes
1answer
21 views

Proving set function is measure

Let $(X,\Sigma,\mu)$ be a measure space. For each $E \in \Sigma$ we define $$\mu_0(E)=\sup\{\mu(F): F \in \Sigma, F \subset E, \mu(F)<\infty\}$$ I got stuck trying to show $\sigma$-additivity, ...
3
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2answers
49 views

Confusion about Lusin's Theorem.

I saw a proof which heavily relied on Lusin's Theorem recently, and I was hoping someone might be able to help me fill in the detail as to why this theorem allows for a particular creation. ...
1
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1answer
35 views

How many infinite subsets of the Naturals have natural density (asymptotic density) zero?

Are there countably or uncountably many? I know that the set of all primes has density zero. Is there an obvious way of using that result to construct an uncountable family of such sets?
1
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1answer
35 views

Prove a collection is a $\sigma$-algebra

I have to prove that a collection of sets is a $\sigma$-algebra. I'm stuck with the axiom of closure under countable unions. The collection is $$ \mathcal{A}=\{A\in\mathcal{B}:m(A\Delta T^{-1}A)=0\} ...
0
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1answer
29 views

Why every set of positive measure has non-measurable subsets

Theorem: If $A \subset \mathbb R$ and every subset of $A$ is Lebesgue measurable then $m(A)=0$ Corollary: Every set of positive measure has non-measurable subsets $m$ in here denote Lebesgue ...
4
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1answer
86 views

Prove $\{ (x,y) \in [0,1]^2: x-y\in \mathbb{Q}\}$ is measurable.

Let $T:=\{ (x,y) \in [0,1]^2\ :\ x-y\in \mathbb{Q} \}$. Show that $T$ has measure zero, but it meets every set of the form $A \times B$ , where $A$ and $B$ are measurable sets of positive measure ...
1
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1answer
22 views

Prove $\int_X |f|^p=p\int^{\infty}_{0} t^{p-1}\mu({x: |f(x)>t}) dt\,$

Let $(X,\mathcal{M},\mu)$ be a measure space and $f$ be a nonnegative measurable function on $X$. Let $1\le p<\infty$. Show that, the function $|f|^p$ is integrable with respect to $\mu$ ...
1
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1answer
29 views

How do I show $P(\mathbb Q) = \mathscr B(\mathbb R) \cap \mathbb Q$

My first question is to confirm that $\sigma$-algebra generated by the set $\{ (a,b] \cap \mathbb Q \mid -\infty \leq a \lt b \leq \infty \}=\mathscr B(\mathbb R) \cap \mathbb Q$ where $\mathscr B$ ...
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0answers
22 views

Proving a reflection principle in probability [closed]

This is a problem that I am stuck at. I tried to prove these exercises by dividing cases, but it only seems to complicate the matter. Could anyone help me how to solve this problem?
0
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1answer
43 views

Any positive measure subset of $\mathbb R$ contains a positive measure Cantor set

A question asks to show any positive measure subset of $\mathbb R$ contains a positive measure Cantor set. How to start with this? I have been staring on this for a while, but can not come up with any ...
3
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2answers
97 views

Can anyone clarify why this is?

The question is to prove: $$I=\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}$$ Let $$I_r=\int_{-r}^{r}e^{-x^2}dx \implies I_{r^2}=\int_{-r}^{r}e^{-x^2}dx\int_{-r}^{r}e^{-y^2}dy=\iint_{[-r,r]^2} ...
1
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0answers
28 views

Partitioning a space such that a set of ergodic measures is uniquely supported on one of the sets

I'm reading a book on Ergodic theory, and it says that given a set X with a sigma algebra A, and a measurable automorphism T, then you can take a set of ergodic measures $\mu_i$ $\epsilon$ M$_T(X)$ ...
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0answers
24 views

Lebegue measure of the union of measurable sets

Let us call a subset $E$ of $[0,1]$ Lebesgue measurable if $\lambda^*(E) + \lambda^*([0,1]\setminus E) = 1$, where $\lambda^*$ is the outer measure. How can we derive from this definition the fact ...
1
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1answer
16 views

Construct measures on $\sigma(B)$ that agree on $B$

Let $X=\{ 1,2,3,4\}$ and $\mathcal B=\{\{1,2 \},\{ 1,3\},\{ 2,4\},\{ 3,4\} \}$. And let $\mathscr A = \sigma(\mathcal B)$ be the $\sigma$-algebra generated by the set $\mathcal B$. I wish to construct ...
0
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2answers
57 views

Supported function within $[a,b]$

Suppose $\phi_n : \mathbb{R} \to \mathbb{R}$ and $|\phi_n(x)| \le 1$ for all $x \in \mathbb{R}$. Also suppose that $\text{supp } \phi_n \subset [a,b]$, with $a,b \in \mathbb{R}$ s.t. $a > b$, ...
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2answers
20 views

Example that fail to be $\lambda$-measurable

I am looking at this definition: Given an outer measure $\lambda$ on $X$, we called a subset $A$ in $X$ $\lambda$-measurable if for every subset $B \subset X$, $$\lambda(B)=\lambda(B \cap A) + ...
1
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1answer
41 views

$L^p$ space as a vector space

I know that the set $L^p(\Omega) = \{f:\Omega \rightarrow R : (\int_\Omega |f|^p)^{1/p}<\infty\}$, where $\Omega \subset R^n$, is a normed vector space. However, if I let $f$ be an extended ...
5
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0answers
29 views

Showing that $f_n\to f$ a.e. $\implies f_n\to f$ almost uniformly where $|f_n|\leq g\in L_1$ [duplicate]

The following problem is from Carothers' Real Analysis: Suppose $f_n$ is a measurable sequence of functions such that $|f_n|\leq g\in L_1$ for all $n$. Prove that $f_n\to f$ almost everywhere ...
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0answers
28 views

Homology group - rational number [closed]

Determine the structure of the homology group H0(X), n>0, if X is the set of rational numbers with their usual topology. This is a question in ''A Basic Course in Algebraic Topology'', William S. ...
1
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1answer
20 views

Relation between vague convergence and weak convergence

This is the Portemanteau Theorem. And this is its corollary. I tried to prove that (i) implies (ii) in this corollary using the Portemanteau Theorem above. But I have kept failed... What is so ...
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0answers
50 views

why( in measure theory ) zero multiply infinite is zero?

why( in measure theory ) zero multiply infinite is zero? I think because the area of line in the plane is zero then zero multiply infinite is zero,is it true?
2
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3answers
28 views

Borel sigma-algebra over [0,1]

I just started studying this, so forgive me if I get something wrong. I have been given the following definition of a Borel $\sigma$-algebra over $\Omega=[0,1]$: It is the smallest $\sigma$-algebra ...
1
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0answers
73 views

Representation of symmetric functions

Let $n \in \mathbb{N}$. Show that every symmetric function $f\colon E^n \rightarrow \mathbb{R}$ can be written in the form $f(x) = g\Bigl(\frac{1}{n}\sum_{i=1}^n \delta_{x_i} \Bigr)$, where $g$ has ...
1
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1answer
17 views

Convergence in distributiom

I need to show that for arbitrary random variables $X_n$, there exist a sequence of positive constants $a_n$ such that $a_nX_n\overset{D}\rightarrow 0$. Thus, I need to show that $\lim_{n\rightarrow ...
0
votes
0answers
23 views

Show $\mathscr B(\mathbb R) \otimes \mathscr B(\mathbb R) = \mathscr B(\mathbb R^2)$

How do I show $\mathscr B(\mathbb R) \otimes \mathscr B(\mathbb R) = \mathscr B(\mathbb R^2)$ where $\mathscr B$ is Borel Algebra. I don't quite sure how to prove this relation, do I show that they ...
2
votes
0answers
49 views

What do we call $\mathfrak{M}$?

I am starting to learn some measure theory, and I was wondering if there is a name for $\mathfrak{M}$. I have the definition: A collection $\mathfrak M$ of subsets of a set $X$ is said to be a ...
2
votes
1answer
23 views

algebra generated by finite set

Is algebra generated by a finite set $A$ same a the $\sigma$-algebra generated by the same set $A$? For example: $X=\{1,2,3,4\}$, $A=\{\{1,2\},\{ 2,3\},\{ 4\} \}$, what is the algebra generated by ...