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

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5
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1answer
72 views

Question regarding Radon-Nikodym derivative…

The problems are as follows: (1) Let $X=[0,1]$ with Lebesuge measure and consider probability measures $\nu,\mu$ given by densities $f,g$ as follows: $$\nu(E)=\int_{E} ...
2
votes
2answers
40 views

Sets in product $\sigma$-algebras that cannot be written as a product of measurable sets in the factors.

I am aware that not every set in a product $\sigma$-algebra can be represented as a product of measurable sets in the factors (e.g., take the unit ball in $\mathbb{R}^{n}$), but this seems weird to ...
4
votes
0answers
27 views

Infinite products of non-measurable sets

I just proved for a homework problem that the direct product of two non-measurable sets is non-measurable. It seems to me that the finite direct product of finitely many non-measurable sets is also ...
4
votes
0answers
46 views

The “muscle” behind the fact that ergodic measures are mutually singular

This is really motivated by the soft question at the end, but let me begin with something more circumscribed: Let $(X,\mathcal{B})$ be a measurable space and let $T:X\circlearrowleft$ be a self-map ...
1
vote
1answer
43 views

Is this c the same as that c?

Are the highlighted $c$'s the same or should it be $c_1$ and $c_2$.
3
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3answers
47 views

Why does the support of measure on $\mathbb{R}^n$ exist?

DEFINITION : The support of a measure on $\mathbb{R}^n$, written spt $\mu$, is the smallest closed set such that $\mu(\mathbb{R}^n \setminus X)=0$. Why does this set exist?
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0answers
16 views

Haar measure of an angle-distance ball in SO3

If for rotations $R_0$, $R_1$ we define the distance $d(R_0, R_1)$ to be the angle of $R_0 R_1^{-1}$ and given $r\in [0,\pi)$, what is the "volume" (normalised Haar measure) in $SO_3$ of the ball ...
0
votes
1answer
37 views

Example of a sequence in L1 with these conditions

Is there an example of a sequence $\{f_n\}$ in $L^1(\mathbb{R})$, such that: $\{||f_n||_1\}$ is bounded. There's a convergent subsequence $f_{\phi(n)}$, i.e. $\exists f \in L^1(\mathbb{R})$ such ...
1
vote
1answer
46 views

Absolutely continuous but not monotone

I don't want to comment on an old question, so I'm asking a new one. The question I'm referring to is Absolutely Continuous and Strictly Increasing on a Subinterval. Specifically, I'm concerned about ...
0
votes
1answer
31 views

Question 4.R of Bartle's Elements of Integration.

Can you help me please? If $f \in M^{+}(X, \mathbf{X})$ and $$\int f d\mu \lt +\infty,$$ then the set $N=\{x \in X: f(x)\gt 0\}$ is $\sigma$-finite (that is, there exists a sequence $(F_n)$ in ...
4
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0answers
44 views

I want to proof that the lebesgue measure of the set below is positiva. Help me!

Let $\Omega$ be a domain limited with smooth boundary in $\mathbb{R}^{n}$ and consider the Sobolev space $H^{1}_{0}(\Omega)$ equiped with the norm $||u||=\int_{\Omega}|\nabla u|^{2}dx$. Let ...
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0answers
14 views

Absolute continuity and signed measures

Suppose that $\nu$ is a signed measure and $\mu$ is a positive measure on $(X, \mathcal{M})$. $\nu$ is $\textbf{absolutely continuous}$ with respect to $\mu$, if $\nu(E)=0$ for every $E \in ...
0
votes
1answer
37 views

measure of open set with measure Haar

By a Haar measure on a locall compact group (Hausdorff) we mean a positive measure $\mu$ (contains the borel set's) such that The measure $\mu$ is left invariant The measure μ is finite on every ...
2
votes
0answers
61 views

Basic Fourier analysis explanation needed wrt a function $f$ and a finite Borel measure $\mu$

An extract from Chapter 12 of Matilla's Geometry of Sets and Measure on Euclidean Spaces I do not believe that formulas (12.1-12.3) are easily seen to be valid. I do not understand what ...
5
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0answers
34 views

Show that the norm $\| \dot\ \|$ and $\| \dot\ \|^{-1}$ preserve Lebesgue measurability.

Let $\| \dot\ \|:\mathbb{R}^n \to \mathbb{R}$ be the euclidean norm. By continuity $\| \dot\ \|^{-1}$ preserves Borel measurability so it suffices to check that it preserves null sets. In the case ...
1
vote
1answer
33 views

$\mathcal L^{\infty}$ space properties

Can anybody give an example that for $1 \leq p < \infty$ neither $\mathcal L^p (\mathbb R) \subseteq \mathcal L^{\infty} (\mathbb R)$ nor $\mathcal L^{\infty} (\mathbb R) \subseteq \mathcal L^p ...
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vote
0answers
24 views

Independent events in “coin tossing space”

This question really puzzles me. The setup: We define the "coin tossing space" by starting with the set $\Omega = \{ 0,1 \}^{\mathbb N}$ and then defining the finitely determined events $t \in ...
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0answers
20 views

Which is a good book to read about convergence of posterior measure?

I am working on Bayesian statistics and would like to know about a good text book about convergence of posterior measure.
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1answer
17 views

Properties of function on $L_p$ spaces

Given $L_p$ space with the lebesgue measure on $\mathbb{R}^n$ and the function $f(x) = |x|^{-\alpha}$ if $|x| < 1$ $f(x) = 0$ if $|x| \geq 1$ I need to show that $f \in L_p$ if and only if ...
0
votes
1answer
35 views

Help in proving $f \circ \phi \in \mathcal L^1(\lambda) \iff \int_0^{\infty} \frac {f(x)}{\sqrt x} \lambda (dx) < \infty$

Consider the measure space $(\mathbb R, \mathcal B(\mathbb R), \lambda)$ and let $\phi: \mathbb R \rightarrow \mathbb R$ be given by $\phi(x) = x^2$. I want to show that for $f \in \mathcal ...
2
votes
1answer
48 views

A basic measure theory question on lebesgue integral

Let $\mu$ and $\nu$ are probability measures on a complete separable space $S$. Suppose, for every real-valued continuous function on $S$ we have that $$\int fd\mu = \int fd\nu$$ does it imply $\mu = ...
1
vote
1answer
24 views

Is the average of a dense orbit ergodic for shift function?

Let $\sigma$ be the shift function in the space of two-sided infinite sequences of $\{0,1\}$, $X=\{0,1\}^\mathbb{Z}$ equipped with product topology. We know that there is some point $x\in X$ with ...
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vote
0answers
28 views

Why is $g(X)$ measurable with respect to $\sigma(X)$

Suppose $X,Y$ independent random variables and $\phi$ be a function such that $E[\phi(X,Y)]<\infty$. Let $g(x)=E[\phi(x,Y)]$. We need to show that $g(X)=E[\phi(X,Y)|X]$. However I cant show ...
0
votes
1answer
18 views

Continuos functions in L_p

The collection of all continuos complex functions on $X$ whose support is compct is denoted by $C_c(X)$. In Rudin Book, Real and Complex Analysis, page 69. Theorem 3.14 For $1\leq p< \infty$, ...
4
votes
1answer
82 views

Show that there is a continuous $g$ with compact support

If $f$ is a measurable complex function (that means that it doesn't take the values $\pm \infty$) with compact support, then for each $\epsilon >0$ there is a continuous $g$ with compact support so ...
0
votes
1answer
35 views

A simple function equals $0$

I'd like to show that if $\mu(A_{k})=0$ then $h=\sum c_j*\chi_{A_{j}} =0$. I can assume that $A_i\cap A_j=\emptyset$ for $i\neq j$. Because I can always make the sets disjoint. What should be my ...
0
votes
1answer
19 views

Compute limit of integral

I'm having trouble with the following question: Compute $ \lim \int_{0}^{1} f_n(x)$ where $f_n(x) = \frac{n x \log x}{1 + n^2 x^2}$ Could I have a hint please? Thank you!
0
votes
1answer
37 views

If $\lim_{n \to \infty} f_n=f$ (Almost everywhere) then $\lim_{n \to \infty} f_n=f$ ( in measure on$E$)

Suppose $E$ is measurable subset of $\Bbb R$, $(f_n)$ is sequence of measurable functions from $E$ to $[-\infty, \infty] $ , $f$ is function from $E$ to $[-\infty , \infty]$. If $\lim_{n \to ...
3
votes
1answer
60 views

Question on $L_p$ spaces involving $\lambda^n$-measure on $\mathbb{R}^n$

Q/ Consider $L_p=L_p(\lambda^n)$ with the Lebesgue measure on $\mathbb{R}^n$ and $1\leq p<\infty$. Let $f_0=|x|^{-\alpha}$ for $|x|<1$ and $0$ otherwise. Show $f_0\in L_p$ iff $p\alpha < n$. ...
2
votes
3answers
98 views

To find a measurable subset with arbitraray measure

Suppose $E$ is measurable subset of $\Bbb R$ s.t. $m(E)=1$ . Is exists $A$ that is measurable subset of $E$ and $m(A)=\frac 1 2$? $A\subset E$ so $m(A) \le m(E)=1$ . since $0 \le m(A)\le 1$, ...
6
votes
1answer
157 views

Class of Lebesgue-Lebesgue measurable functions?

A function $f:\mathbb{R}^n\to\mathbb{R}^m$ is Lebesgue-Lebesgue measurable iff inverse images of Lebesgue measurable sets are Lebesgue measurable. Seems to me that since projections* and arithmetic ...
1
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3answers
28 views

Chain of Implications for Continuity and Boundedness

Consider the following definitions: > 1). Somewhere Locally Bounded: $\exists p \in X, \exists \epsilon >0, \exists \delta >0, \forall q \in X: d(p,q)< \delta \Rightarrow ...
1
vote
1answer
29 views

Proving that the Bernoulli self similar measure is doubling

Let $\mu_p$ a measure which is the push forward of the bernouli product measure $(p,1-p)^\mathbb N$. Let S=$\{f_1,\dots f_m\}$ an IFS, a system of functions with attractor $K$, means ...
4
votes
1answer
69 views

Rules for translating quantifiers to set operations?

I had this excercise in measure theory where I had to show that certain sets are measurable and I realized there was some mechanical procedure going on. Here is the question: Let $f_n:X\to ...
4
votes
1answer
78 views

Computing an explicit Radon-Nikodym derivative

Q/ let $\lambda$ be the Lebesgue measure and $\delta_0$ be the Dirac measure at 0. Show that $\lambda$ is abs cts wrt $\lambda+\delta_0$ (have done this part) and find the R-N derivative ...
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0answers
26 views

Investigate the existance of the following integrals (in a measure theory context)

I've been asked to investigate the existence and equivalence of the these integrals: $$\int^0_1\int^0_1f(x,y)d\lambda(x)d\lambda(y)\text{ and }$$ $$\int^0_1\int^0_1f(x,y)d\lambda(y)d\lambda(x)$$ (yes ...
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1answer
57 views

Question on $L_p$ spaces

Consider $L_p = L_p(\lambda^n)$ with the Lebesque measure on $\mathbb{R}^n$ and $1 \leq p < \infty$. Let $f_0(x) = |x|^{-\alpha}$ if $|x| < 1, f_{0}(x) = 0$ for $|x| \geq 1$. Show that: $f_{0} ...
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0answers
14 views

Characterisation of Jordan measurability

If we define finite union of boxes as elementary sets. Then define a measure of these sets as follows: Let $E \subset \mathbb{R}^d$,If $E$ is partitioned as the finite union $B_1 \cup \cdots \cup ...
2
votes
1answer
32 views

Measurable in functions (equal state)

Let $f: \Bbb R\rightarrow \Bbb R $ with this properties: $\forall \epsilon >0$ , $\exists U\subset\Bbb R$ , $U$ is open set, s.t.$m(U)<\epsilon$ , $f$ is continuous on $ \Bbb R$\ $U$ ...
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0answers
28 views

Bochner Integral: Measurability

Problem Given a measure space $\Omega$ and a Banach space $E$. Consider a Bochner measurable function $S_n\to F$. Then it admits an approximation from nearly below: $$\|S_n(\omega)\|\leq ...
2
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1answer
23 views

Comparing lebesgue measure and counting measure

I have the following problem from Folland: Let $X = [0, 1]$, $\mathcal{M} = \mathcal{B}_{[0, 1]}$, $m = $ Lebesgue measure and $\mu = $ counting measure. $m \ll \mu$ but $dm \neq f \, ...
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0answers
22 views

Integrating a function of measures

I've been reading John Baez's series of posts on Information Geometry. I'm currently on part 6... Midway through the post he discusses Radon-Nikodym derivatives: The formula for information gain ...
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0answers
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Understanding a proof of Lusin. Question about measure of the set discontinuity point of a step function defined on finite measurable sets.

I'm reading a proof and I don't understand the red part. I think that if $f_n$ is a step function, then the set of points in which $f_n$ is discontinuous is countable, in particular has $0$ lebesgue ...
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1answer
100 views

Stone's Theorem Integral: Bad Example!

Disclaimer This thread is just to record. See: Answer own Question It is stated as question for jeopardy. Have fun. :) Problem Given a finite Borel measure $\mu(\mathbb{R})<\infty$ and a Banach ...
2
votes
2answers
38 views

a relation between $f$ ,$|f|$ in measurability

Suppose $f: \Bbb R \rightarrow \Bbb R $ is function s.t. $|f|$ is measurable Is the $f$ measurable? (True or False) . if $|f|$ is measurable and $\alpha $ is arbitrary then $\{ x \in \Bbb ...
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votes
3answers
280 views

measurability with zero measure

Let $f : [0,1] \rightarrow \Bbb R$ is arbitrary function , and $E \subset \{ x \in [0,1] | f'(x)$ exists$\}$. How to prove this statement: If $E$ is measurable with zero measure then $f(E)$ ...
3
votes
1answer
27 views

Instance where local convergence in measure implies global convergence in measure

Let $(X, \mathscr{A},\mu)$ be a measure space. Let $\gamma\subset L^0$ and suppose that for each $\epsilon>0$ and $\epsilon'>0$, there exists an $A\in\mathscr{A}$ with $\mu(A)<\infty$ such ...
1
vote
1answer
12 views

How to calculate the closeness of a set of numbers?

Given a set of numbers, I would like to have a measure of how close they are to each other. I would like the calculated measure produces a single value. How could I achieve this?
2
votes
2answers
55 views

Show $\|f-1_{[0,1/2]}\|_{\infty} \geq 1/2$ for any continuous function $f:[0,1] \to \mathbb{R}$

Show that for any continuous map $f$ on $[0,1]$, the indicator map on [$0$,$1/2$] and $f$ has norm difference at least $1/2$. I have been trying to prove it using the intermediate value property of ...
2
votes
1answer
34 views

Lebesgue Integral of function multiplied by infinity

In Rudin's Real and Complex Analysis there is a following proposition: 1.24 Let $f$ be measurable function, $E$ measurable set, and $\mu$ a measure. Then if $f \geqslant 0$ and $c$ is a ...