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

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X is some random variable and f is a continuous function. Is f[E(X)] = E[f(X)]?

I am curious about at what conditions the expectation and a mapping could exchange their operation. Say, X is some random variable, and $f:R\rightarrow R$ is a continuous function. Does $$f[E(X)] = ...
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1answer
53 views

integral of lebesgue function is continuous

Let F be a lebesgue integrable function on $(0,\infty)$. For $0 \le t < \infty$, define $g(t)=\int_{0}^{\infty} e^{-tx}F(x)dx$. Can someone explain why $g$ and $g'$ are continuous over ...
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0answers
29 views

A series of measures

Let $(\mu_n)$ be a sequence of finite measures on $(\Omega,\Sigma)$. We put: $$\mu(A):= \sum\limits_{n=1}^\infty \frac{\mu_n(A)}{2^n(1+\mu_n(\Omega))}$$ Show that $\mu$ is a finite measure. I have ...
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0answers
14 views

Definition of an “arc” and possible error in proof of length of projection of regular $1$-set in $\mathbb{R}^2$.

Here is an extract from Falconer's The Geometry of Fractal Sets. I cannot see how an "arc" is defined and was wondering whether someone could help me with the definition. Also if $ \begin{align} ...
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1answer
15 views

What is meant with translation-invariant product- measure here?

Consider $X=\left\{0,1,2\right\}^{\mathbb{Z}^2}$. What is meant by a tranlation-invariant product-measure on $X$? On which $\sigma$-algebra? What does translation-invariant mean here?
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36 views

How is this passage probably meant?

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}^d}$. Let $\mathfrak{B}$ denote the Borel field on $X$ generated by its topology and let $\mu_{p_0,p_1,p_2}$ be product measure on $X$ in which the $i$'s have ...
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1answer
32 views

Convergence in measure of sequence of functions

Hi I don't have a lot of experience in measure theory so that might be basic. If you have a sequence of functions $a_{k}(x): \Omega \rightarrow \mathbb{R}$ such that $$0 \leq\limsup\limits_{k ...
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1answer
51 views

Evaluation of Lebesgue Integral using Convergence Theorems

Using convergence theorems, I am trying to compute the value of $$ \lim_{n\to\infty}\int_a^\infty \frac n{1+n^2x^2}\,\mathbb{d}x $$ for $a \in \mathbb{R}$, and with respect to the Lebesgue measure. ...
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1answer
23 views

Bochner Integral: Approximability

Disclaimer This thread is related to: Bochner Integral: Integrability It is meant to record. See: Answer own Question It is written as jeopardy. Have fun! :) Problem Given a measure space ...
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1answer
25 views

Showing that $||\hat{f}||_{\infty} \leq ||f||_1$ in $L^1$

Let $f \in L^1(\mathbb{R}^n)$ then $\hat{f} \in L^{\infty}(\mathbb{R}^n)$ and $||\hat{f}||_{\infty} \leq ||f||_1$ How do you prove this or where can I find a proof of this fact?
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What is general relationship between Lebesgue-Stieltjes measurability and Lebesgue measurability?

What is general relationship between Lebesgue-Stieltjes measurability and Lebesgue measurability? I know Borel measurability implies both Lebesgue-Stieltjes measurability and Lebesgue measurability. ...
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2answers
80 views

Amann & Escher Integral vs. Lebesgue Integral

In the textbook the authors define the integral via cauchy sequences of simple functions: $$S_n\to F:\quad\int F\mathrm{d}\mu:=\lim_n\int ...
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1answer
73 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} ...
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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 ...
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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 ...
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47 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 ...
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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 $c_2$.
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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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 ...
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1answer
36 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 ...
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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 = ...
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1answer
25 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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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 ...
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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 $1\leq p< \infty$, ...
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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 ...
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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 ...
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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!
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1answer
38 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 ...
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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 $p\alpha < n$. ...
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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 $0 \le m(A)\le 1$, ...
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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 ...
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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 ...
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1answer
30 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 ...
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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
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1answer
79 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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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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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 ...