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

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What is a sequence of functions that converge weakly in Lp, but not strongly?

I am reading Royden’s real analysis. In his book, a sequence of functions in Lp converges weakly if every bounded linear functional in the dual space converges in R. Can anyone discuss the ...
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0answers
23 views

Show that $f$ defined on the interval $(a,b)$ is not differentiable for every point in $E$ with $m(E)=0$

Let $E$ have measure zero contained in the open interval $(a,b)$. In a previous problem I showed that there is a countable collection of open intervals, $\{(c_k,d_k)\}_k$, contained in $(a,b)$ for ...
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1answer
28 views

If a sequence converges in measure, are convergent subsequences of it all converge to the same limit?

Let $f_n:X\rightarrow \mathbb{C}$ be a sequence of measurable functions such that $f_n\rightarrow f$ in measure. Let $f_{n_k}$ be a subsequence of $f_n$ such that $f_{n_k}\rightarrow g$ pointwise a.e....
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1answer
24 views

Jacobian for $X = YX'$ where $X',Y,X$ are $n\times n$ matrices?

I'm trying to work through this example on the wiki for Haar measures, showing that $$ \mu(S) = \int_S \frac{1}{|\det(X)|^n}\,dX $$ is a left Haar measure for $\mathrm{GL}(n,\mathbb{R})\subseteq \...
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1answer
43 views

information measure for matrix that is analogous to rank

Is there a measure for matrix that is analogous to rank of the matrix, but it is continuous on matrix elements? Say, we could say the information in identity matrix $I_n$ is $n$, and when the off-...
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1answer
26 views

Variation of Egoroff's theorem.

Let $(X,\mu)$ be a measure space with a (positive) measure. Let $\{f_n : X \to \mathbb{R} : n =1 ,2,...\}$ be a sequence of measurable functions satisfy the following properties : For each $n\in \...
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0answers
19 views

Change of Variables for Integral over a Fractal

We define the integral of a function $f(x,y)$ over a fractal $F$ to be, $$(1) \quad \int_F f(x,y) \ d\mu(x,y)$$ Where $\mu$ is the normalized Hausdorff measure. Expressed another way, we have, $$\...
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1answer
40 views

Baire measurable sets

I got the following setting: Let $X$ a uncountable set, $p \notin X$ and $X^* = X \cup \{p\}$. Let $$\mathcal O := \{O \subseteq X^* \mid O \subseteq X \text{ or } p \in O \text{ and } X \setminus O \...
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1answer
26 views

Is there a meaningful measure on analytic functions?

Let $\mathcal{B}$ be the functions analytic on the unit disk and continuous on its boundary. With the supremum norm this becomes a Banach space. Is there any way to define a meaningful measure on ...
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17 views

Example of linear functional from Rudin's book

Rudin states that $\Lambda(f)=\int \limits_{X}fg d\mu$ is linear functional. We know that $\exists M$ such that $|g|\leqslant M$. Hence $|fg|\leqslant M|f|$ $\Rightarrow$ $\int \limits_{X}|fg|d\mu<\...
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1answer
20 views

Definition of complex vector space from Rudin RCA

This is definition of complex vector space from Rudin's book. He write that to each pair $(\alpha,x)$, where $x\in V$ and $\alpha$ is scalar there is associated a vector $\alpha x\in V$. That's right. ...
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1answer
61 views

Uniqueness of the uniform spherical distribution

Suppose that $X,Y$ are random vectors on some (possibly different) probability spaces mapping to $\mathbb R^n$ for some $n\in\mathbb N$. Suppose furthermore that $\|X\|=r>0$ for all realizations ...
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\...
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1answer
77 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 ...
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1answer
45 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 $\int f_n\,d\lambda =1$, ...
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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 ...
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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 $$\int \limits_{X}cfd\mu=c\int \limits_{X}fd\mu \qquad (*)$$ ...
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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 ...
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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.
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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 ...
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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 ...
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1answer
38 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 $A$...
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1answer
28 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 ...
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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 ...
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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
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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 ...
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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 $\...
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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 ...
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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 ...
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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 \...
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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}...
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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) < \...
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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.) $E\in M_{\mu}$ ...
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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 ...
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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 \...
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1answer
119 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>...
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1answer
37 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 ...
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0answers
138 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 ...
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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 ...
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1answer
21 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 ...
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2answers
60 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
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1answer
51 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 ...
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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. ...
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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 ...
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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) &...
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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
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1answer
40 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
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2answers
46 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} \...
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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 \...