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

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2
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1answer
61 views

If $A$ is measurable, is $TA, T\in\text{End}(\mathbb{R}^n)$ measurable?

Let us define, as Kolmogorov-Fomin's Элементы теории функций и функционального анализа does, the definition of outer measure of a bounded set $A\subset \mathbb{R}^n$ as $$\mu^{\ast}(A):=\inf_{A\...
5
votes
1answer
70 views

Why are there no finitely additive measures on $\ell_\infty$ for which the measure of every ball is positive and finite?

As the question title suggests, why are there no finitely additive measures on $\ell_\infty$ for which the measure of every ball is positive and finite? Here, we do not assume that the measure is ...
1
vote
1answer
44 views

Where $\{q_n\}=\mathbb Q$ and $f_n:[q_n-2^{-n-1},q_n+2^{-n-1}]\to[0,\infty)$ with $\int f_n\,d\lambda=1$, show $\sum_{n=1}^\infty f_n<\infty$ a.e.

That is: Let $\mathbb Q=\{q_n\}_{n\in\mathbb N}$ be an enumeration of the rationals. Let $f_n$ be a nonnegative Borel measurable function supported on $q_n\pm 2^{-n-1}$ with $\int f_n\,d\lambda =1$, ...
1
vote
2answers
48 views

Real Analysis, Folland Theorem 1.21 Borel Measures

Background information: $L$ is the class of Lebesgue measurable sets. $m$ is the Lebesgue measure which is a complete measure $\mu_F$ associated to the function $F(x) = x$, for which the measure of an ...
0
votes
1answer
30 views

$\int \limits_{X}cfd\mu=c\int \limits_{X}fd\mu$ if $c=\infty$ and $f\geqslant 0$

Let $(X,\mathfrak{M},\mu)$ - measure space and $E\in \mathfrak{M}$ and $f:X\to [0,\infty]$-measurable function and $c=\infty$. Prove that $$\int \limits_{X}cfd\mu=c\int \limits_{X}fd\mu \qquad (*)$$ ...
0
votes
2answers
25 views

Sigma-algebra clarification

Suppose $F$ is a sigma-algebra, $A\subset B$, $B\in F$. Is it the case that $A\in F$? I'm familiar with the definition of a sigma-algebra (closed under complements and countable unions and ...
0
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0answers
14 views

Can anyone help me understand one step in the proof related to weak convergence in Lp space?

Can anyone explain where equation (19) is from in the pasted page below? This is from Royden chapter 8 thm 7 on page 164.
3
votes
1answer
38 views

Change of Variable Proof in Folland

I am reviewing Folland's proof of the following standard result and I have a question on one part. Suppose $\Omega$ is an open set in $\mathbb R^{n}$; $G:\Omega \to \mathbb R^{n}$ is a diffeomorphism ...
2
votes
1answer
44 views

Another question about proving Lebesgue Decomposition

Note: This is my original question. I have been kindly helped to turn this into a correct proof, which I have posted as an answer so this question won't show up as "unanswered". As an exercise, I am ...
0
votes
1answer
34 views

Lipschitz transformation maps measure zero sets to measure zero sets. [duplicate]

Let $T:\mathbb{R^2} \to \mathbb{R^2}$ be Lipschitz function. Then, (a) If $E$ is a set in $\mathbb{R}^2$ with Lebesgue measure zero, then $T(E)$ has measure zero in $\mathbb{R}^2$. (B) If $A$...
1
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1answer
27 views

Reference request: product Borel $\sigma$-algebra of non-separable metric spaces

The following is a proposition in Folland's Real Analysis about product sigma algebra: Here $\mathcal{B}_X$ denotes the Borel $\sigma$-algebra on $X$. Could anyone come up with an example that ...
1
vote
1answer
38 views

An exemple of strict inequality for reverse inequality Minkowski for space $L^p$, $0 < p <1$

Let be $0<p<1$. Suppose that we know that $$ \bigg(\int (u + v)^p\bigg)^{1/p} \geq \bigg(\int (u)^p\bigg)^{1/p} +\bigg(\int (v)^p\bigg)^{1/p}$$ for all $u,v \geq 0$ in $L^p$. I need find an ...
1
vote
2answers
31 views

Does a set of positive measure in a product $\sigma$-algebra contain a rectangular set

Suppose $(E_i, \mathcal E_i)$, $i = 1, \dots, n$, are measurable spaces and let $E := E_1 \times \dots \times E_n$, equipped with the product $\sigma$-algebra, denoted by $\mathcal E$. Suppose $\psi$ ...
0
votes
0answers
12 views

Let $Q=[0,1]\times[0,2]$. Find the rotation matrix $(A)$for the angle $\frac{\pi}{4}$ upon this set. Is $A(Q)$ measurable and a elementary set?

Let $Q=[0,1]\times[0,2]$. Find the rotation matrix $(A)$for the angle $\frac{\pi}{4}$ upon this set. Is $A(Q)$ measurable and a elementary set? First off, I know that $A$ is a linear map, and a ...
-3
votes
1answer
37 views

intersection of two set of positive measure has positive measure [closed]

Suppose we have $\Omega_1 \subset \Omega,\Omega_2\subset\Omega$ such that \begin{equation} \mathbb{P}(\Omega_1)=1 \qquad \mathbb{P}(\Omega_2)=\varepsilon \in (0,1) \end{equation} How can I show that $\...
0
votes
1answer
44 views

If $\int_0^1 |f|dx= (\int_0^1|f|^p dx)^{1/p}$ for some $p > 1$, then $f$ is constant.

If $\int_0^1 |f|dx= (\int_0^1|f|^p dx)^{1/p}$ for some $p > 1$, then $f$ is constant. Since $\int _0 ^1 |f| dx = \int _\mathbb{R} fg dx $ where $g \mbox{ is characteristic function on } [0,1]$, I ...
0
votes
0answers
14 views

Can I form a measure given the density of a set of natural numbers?

Let $A\subset \mathbb N$. Define $\bar{d}(A)=\limsup_{k\to\infty}\dfrac{|A\cap[1,k]|}{k}$. My question is, can I consider $\bar{d}(A)$ to be a finitely additive measure of $A$? I think no because for ...
1
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0answers
27 views

Problem 10 chapter 1 from RCA Rudin

Suppose $\mu(X)<\infty$, $\{f_n\}$ is a sequence of bounded complex measurable functions on $X$, an $f_n\to f$ uniformly on $X$. Prove that $$\lim \limits_{n\to \infty}\int \limits_{X}f_nd\mu=\int \...
1
vote
0answers
53 views

Extending a probability measure to the sigma field obtained by adjoining a new set

Suppose that $P$ is a probability measure on a sigma field $\mathcal{B}$ and suppose $A\not\in\mathcal{B}$. Let $$\mathcal{B}_{1}=\sigma(\mathcal{B},A)$$ be the sigma algebra generated by $\mathcal{B}...
1
vote
1answer
37 views

Real Analysis Folland, 1.20 Proposition Borel measures

1.20 Proposition - If $E\in M_{\mu}$ and $\mu(E) < \infty$, then for every $\epsilon > 0$ there is a set $A$ that is finite union of open intervals such that $\mu(E \ \triangle \ A) < \...
1
vote
1answer
54 views

Real Analysis, Folland Theorem 1.19 Borel Measures

I made a post about this theorem before. But I decided to create a new post to see if I am proving this theorem correctly. Theorem 1.19 - If $E\subset \mathbb{R}$, TFAE: a.) $E\in M_{\mu}$ ...
0
votes
1answer
58 views

Absolute continuity and continuity

Suppose we have a measure $\mu$ on $(a,b]$ such that $\mu(a,b]=F(b)-F(a)$ where $F$ is non-decreasing, continuous function from the right, Definition: A function $F$ is said to be absolutely ...
2
votes
0answers
34 views

Monotove Convergence theorem for decreasing sequence

Suppose $f_n: X\to [0, \infty]$ is measurable for $n = 1, 2, 3, ...,$ $f_1 \geqslant f_2 \geqslant f_3 \geqslant · · · \geqslant 0,$ $f_n(x) \to f(x)$ as $n\to \infty$, for every $x\in X$, and $f_1 \...
7
votes
1answer
118 views

$\int \limits_{E}|f|d\mu<\varepsilon$ whenever $\mu(E)<\delta$.

Suppose $f\in L^1(\mu)$. Prove that to each $\varepsilon>0$ there exists a $\delta>0$ such that $\int \limits_{E}|f|d\mu<\varepsilon$ whenever $\mu(E)<\delta$. Proof: Let $\varepsilon>...
0
votes
1answer
35 views

Showing an ergodic toral automorphism is not measurably isomorphic to an ergodic circle rotation

The question as listed in the title is the question statement, only I do not want to use that one is mixing and the other is not. Is it true that measurably isomorphic spaces are either both mixing ...
5
votes
0answers
133 views

Demystifying invariant measures in probability theory

Just trying to understand, at least conceptually, invariant measures, and specially their role in probability theory. To be brief: I understand that if we have a set $X,$ with $A \subset X$ just a ...
0
votes
3answers
27 views

show $f_n = \frac{1}{n} \chi_{[n,\infty]}$ is monotonically decreasing

How do I show $f_n = \frac{1}{n} \chi_{[n,\infty]}$ is monotonically decreasing? I know that $\frac{1}{n}$ is monotonically decreasing, but I am unsure how to show $f_n$ is monotonically decreasing ...
1
vote
1answer
20 views

Distributional derivatives for functions that is continuous but nowhere differentiable

It is well known that the Brownian motion is an example of functions that is continuous but nowhere differentiable. In addition, its distributional derivative can be interpreted in the way mentioned ...
-2
votes
2answers
59 views

How to show that the measure of $\cap_n E_n $ is not $0$?

Let $E_n$ ($n \in \mathbb{Z}_{\geq 1}$) be the union of a finite set of closed intervals and the sum of the lengths of the intervals is large than or equal to a fixed positive number $a > 0$. ...
2
votes
1answer
47 views

Upper and lower bound on $L^1$ norm purely in terms of measure

Suppose $f$ is a measurable almost everywhere finite function on $\mathbb{R}^d$, and let$$E_n = \{x : 2^n \le |f(x)| < 2^{n + 1}\}, \quad n \in \mathbb{Z}.$$What is a non-trivial upper and lower ...
2
votes
0answers
40 views

Simple proof of uniqueness of Lebesgue Decomposition?

Lebesgue's Decomposition Thm states: if $\lambda,\mu$ are $\sigma$-finite measures on a measurable space $(X,\textbf{X})$, then $\exists$ unique measures $\lambda_1,\lambda_2$ on $(X,\textbf{X})$ s.t. ...
1
vote
1answer
34 views

For $E\in \Bbb{R}^3$,$(x,y,z)\in E \iff (x,y,-z)\in E$, and linear $ f:\Bbb{R}^3\to \Bbb{R}$, if $(x_0,y_0,z_0)$ is the center of mass, $z_0=0$

Let $E\subset \Bbb{R}^3$ be a measurable set (i.e. $\int_{\Bbb{R}^n}1_{E}$ exists) and let $v(E)\ne 0$. Let $f$ be a linear function $f:\Bbb{R}^3\to \Bbb{R}$, and let $(x_0,y_0,z_0)$ be the center of ...
2
votes
1answer
34 views

Can be summation of numbers in a set considered its measure?

I have for example following sets: \begin{align} A &= \{1,2\}\\ B &= \{3,4\} \end{align} Now I would define my measure as a summation of all the numbers in a set, so: \begin{align} \mu(A) &...
0
votes
0answers
9 views

Regular vs. Borel regular

It is well known we have the notion of regular Borel measure. On the other hand, we know the Lebesgue measure is regular in each measurable set not only on Borel sets. The question is, does there ...
3
votes
1answer
39 views

Give an example of a non-Lebesgue measurable function $f:\mathbb R \to \mathbb R $ such that $|f|$ is a measurable function and …

Give an example of a non-Lebesgue measurable function $f:\mathbb R \to \mathbb R $ such that $|f|$ is a measurable function and $f^{-1}(\{a\})$ is a measurable set for each $a \in \mathbb R$. Can ...
3
votes
1answer
52 views

Limit of integral with measure with parameter $\alpha$

Suppose $\mu$ is a positive masure on $X$, $f:X\to [0,\infty]$ is measurable, $\int \limits_{X}fd\mu=c$, where $0<c<\infty,$ and $\alpha$ is a constant. Prove that $$\lim \limits_{n\to \infty}\...
1
vote
2answers
44 views

Questions on measurable sets

I'm learning about measure theory, specifically measurable sets, and need help with the following exercises: $(1)$ Find the measure of the set $E_1 = \mathbb{Z} \cup \mathbb{Q} \cup (\mathbb{R} \...
1
vote
2answers
49 views

What is the measure of $A=[-1,2]\times[0,3]\times[-2,4]\cup[0,2]\times[1,4]\times[-1,4] \setminus [-1,1]^3$?

I really get stuck after one point, and don't know where to go on.I know that my try, up to where I am stuck is correct. $$\color{#20f}{\text{TRY:}}$$ $$B_1=[-1,2]\times[0,3]\times[-2,4],\mu(B_1)=3 \...
7
votes
1answer
216 views

How to prove that there exists $\lambda_{\sigma(1)}$ such that $\mu(A\cap\{\lambda_{\sigma(1)}\neq0\})>0$?

Let $(\mathcal F,\Omega,\mu)$ be a measure space and $A\subseteq\Omega$ such that $\mu(A)>0$. Let $L^0$ be the space of all measurable functions. We say $X_1,\ldots,X_k\in(L^0)^d=\prod_{k=1}^dL^0$...
0
votes
1answer
43 views

An open set that has no volume

Let $\mathbf Q\cap[0,1]=\{q_n\}_{n=1}^\infty$ and $A=\bigcup_{n=1}^\infty(q_n-\frac1{2^{n+2}},q_n+\frac1{2^{n+2}})$. Show that $A$ has no volume. Here "volume of a set $B$" means the Riemann-...
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0answers
26 views

Improbably doesn´t mean impossible

I have a question: in the past days I give myself the question: given a measurable space $(\Omega,\mathcal{F},\mu)$, what kind of conditions about $\mathcal{F}$ and $\mu$ we need for: The only ...
0
votes
1answer
21 views

a proposition in the construction of the Borel measures on the real line

In the construction of the Borel measures on the real line, the following proposition is used in Folland's Real Analysis: Here is my question: If one replaces $(a_j,b_j]$ with $[a_j,b_j)$, ...
1
vote
1answer
53 views

Two questions on measurable sets.

I'm learning about measure theory, specifically measurable sets, and need help with the following two questions: $(1)$ For $n \in \mathbb{N}$, let $E_n = \{x \in [0, 2\pi] : \sin x < {1 \over n}...
0
votes
1answer
30 views

Show that $\mu(E \cup A) + \mu(E \cap A) = \mu(E) + \mu(A)$

Let $E$ be a measurable subset of $X$, then show that for every subset $A$ of $X$ the following equality holds: $$\mu(E \cup A) + \mu(E \cap A) = \mu(E) + \mu(A).$$ I know since $E$ is measurable $$\...
2
votes
0answers
38 views

Integration w. r. t. counting measure

I'm learning about measure theory, specifically integration w.r.t. counting measure, and need help to verify my understanding of this new notion through two exercises. (1) Let $(\mathbb{N},\scr{P}...
0
votes
1answer
24 views

How do I show that $\mu$ is a measure?

Let $S$ be a semiring, and let $\mu: S \to [0,\infty]$ be a set function such that $\mu(A) < \infty$ for some $A \in S$. If $\mu$ is $\sigma$-additive, then show that $\mu$ is a measure. What ...