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

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2answers
35 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 ...
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0answers
15 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 ...
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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 ...
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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 ...
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1answer
21 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
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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, ...
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2answers
50 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. ...
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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?
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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\} ...
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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
24 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
32 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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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
98 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} ...
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0answers
30 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
26 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
58 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 ...
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0answers
31 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 ...
1
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1answer
21 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?
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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 ...
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0answers
74 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 ...
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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
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0answers
51 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
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1answer
24 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 ...
2
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1answer
38 views

A basic question regarding Lebesgue's density theorem

Here is the question from Pugh's Real Mathematical Analysis: My answer to $b)$ is that for a closed square, points on corner has density $1/4$, while on the sides the density is $1/2$. But how to ...
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1answer
22 views

Proving certain function is a measure

Let $(X,\Sigma)$ a measurable space. Let$\mu:\Sigma \to \mathbb R_{\geq 0}$ be the set function that satisfies: (a)$A,B \in \Sigma$ and $A \cap B=\emptyset \implies \mu(A \cup B)=\mu(A)+\mu(B)$ (b) ...
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1answer
20 views

Cavalieri's Principle in measure theory

The first part of Cavalieri's principle (in measure theory) states if $E$ is measurable, then almost every slice $E_x$ of $E$ is measurable. Here, it uses "almost every", so what is an example where ...
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0answers
22 views

Lebesgue Measurable By Alternative definition of Measure

Prove that any compact set $K$ in $R^{n}$ is Lebesgue measurable and $m(K) < \infty$ Actually the proof of this is given in Stein and Shakarchi's book on Real Analysis (Page 38, Property 4) where ...
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0answers
45 views

If 2-dimensional Lebesgue measure is one then $S_x$ has 1-dimensional measure of one

I am working on a problem$^{(1)}$ like this: Suppose $A$ is a Lebesgue measurable subset of [0, 1]$^2$ with 2-dimensional Lebesgue measure $m_2 (A) = 1$. Show that for almost every $x \in [0, 1]$ ...
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1answer
39 views

Prove $\cos(n_k x)$ does not converge pointwise on $[0,2\pi]$

There are strictly increasing sequence $n_1<n_2<\dots<n_k<\dots$ which are positive integers. I want to prove on the domain $[0,2\pi]$ where $\cos(n_kx)$ converges does not coincide with ...
4
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1answer
126 views

A quite strange question about measure

this strange question just occurred to me, that say $E\subset[0,1]$ and $mE=1$, does it imply the closure $\overline E=[0,1]$? Or is there a closed proper set in $[0,1]$ that has measure $1$? In that ...
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0answers
26 views

Yet another proof of uniquness of Haar measure

I'm trying to prove the uniqueness of Haar measure in my way. Let $G$ be a locally compact Hausdorff group. To simplify matters we assume the underlying topological space of $G$ has a countable base. ...
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0answers
35 views

Prove that $f(x)=e^{-1 /\Vert x \Vert^\alpha}$ is measurable function.

Let $\Omega$ be a bounded domain in $\mathbb{R^N}$ and $f(x)=e^{-1 / \Vert x \Vert^\alpha}$. Prove that $f(x)$ is a measurable function. I wonder if I have to add more condition to $f(x)$. For ...
4
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1answer
72 views

Monotone increasing continuous function with $\int_a^b f' = f(b) - f(a)$ which is not absolutely continuous

If $f:[a, b] \to \mathbb{R}$ is continuous and real-valued, f' integrable on [a, b], and $\int_a^b f' = f(b) - f(a)$, must f be absolutely continuous? What if f is monotone increasing? For the ...
2
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1answer
41 views

Is this statement about measurable function true? [duplicate]

If $f$ is a measurable function on $[0,1]$, then there exists a measurable set $D\subset [0,1]$ such that $mD>0.9$, and a continuous function $g:[0,1]\to R$ such that $f=g$ on $D$. My question is ...
0
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3answers
49 views

$\mathbb{R}^k$ is of measure zero in $\mathbb{R}^l$, $k < l$. [closed]

How do I show that $\mathbb{R}^k$ is of measure zero in $\mathbb{R}^l$, with $k < l$?
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1answer
33 views

A class of functions dense in $L^2$

Suppose $f\in L^2([0,1],\Sigma,\mu)$. Is the class of all $$f=\sum_{i=1}^n \alpha_i (\chi_{A_i}-\chi_{[0,1]/A_i} )$$$A_i\in \Sigma$ to be dense in $L^2([0,1],\Sigma,\mu)$? Thanks.
2
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1answer
40 views

prove that $\int_{\Omega}|f_n-f_0|d\mu\rightarrow 0$ (By weaker assumption on Scheffé's lemma)

I'm dealing with this problem. Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $\{f_n\}$ a sequence of nonnegative integrable functions. Suppose $f_n\xrightarrow{\mu} f_0$ and ...
1
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0answers
41 views

Absolutely continuous iff continuous of bounded variation

I have the following problem (taken from q1 p341 of Kolmogorov and Fomin's Introductory Real Analysis), which I am struggling to prove completely. I think I know how to show the only if part, but not ...
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0answers
23 views

Are $L1$ functions with a.e. finite support a.e. equal to a continuous function?

I was wondering about this: Let $f \in L^1(\Omega)$ and $\Omega\subset \mathbb{R}^n$ be compact, then $f$ is the $L^1$ limit of continuous functions with support in $\Omega$. Egorov's theorem tells us ...
1
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0answers
23 views

continuity at a point in a cadlag process

I am reading a proof that uses the fact: Let $(X_t)_{t \geq 0}$ be a cadlag process. We know that $X(\omega)$ has at most countably many discontinuities, for each $\omega \in \Omega$. It is then ...
2
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0answers
17 views

For $f\in L^1_{loc} (\Omega)$, $f=0$ almost everywhere in $\Omega$ provided $\int_{\Omega}f(x)\Phi (x)dx=0 , \forall \Phi \in C_{c}^{\infty}(\Omega)$

I need to show that $f=0$ almost everywhere in $\Omega$ provided $$\int_{\Omega}f(x)\Phi (x)dx=0 , \forall \Phi \in C_{c}^{\infty}(\Omega)$$ Here is how I have decided to proceed. Suppose there ...