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

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2
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1answer
879 views

An example of a generalized Cantor set with positive Lebesgue measure [duplicate]

I want to know if there exist a set $ X\subset \mathbb R$ such that $X$ is $i)$ Perfect $ii)$ Compact $iii)$ Has empty interior $iv)$ Totally disconnected $v)$ Is not countable But $X$ has ...
2
votes
1answer
122 views

A theorem about the Poisson Point process.

In the proof of the Levy-Khintchine theorem, I saw a theorem about the Poisson point process. The theorem states that if $\Pi$ is a poission point process on $S$ with intensity measure $\mu.$ Let $f:...
2
votes
1answer
68 views

Simplification of an expression

How do I simplify the following expression? $$\displaystyle \frac{\int_q^1 w(s) \int_0^s e(\xi) d\xi ds}{2\int_q^1 w(s) ds} p$$ where $w(t)$ is nondecreasing $w(t)>0$ on $(q,1]$ , $e :(0,1)\...
2
votes
1answer
74 views

Basic question about the definition of an integral on a measure space

Let $(X,\mathcal{B},\mu)$ be a measure space. $\bf{\text{Definition:}}$ For a non-negative measurable function $f$ on $X$, $E\in \mathcal{B}$, $$\int_{E}f d\mu := \text{inf}\int_{E}\varphi d\mu$$ ...
2
votes
1answer
96 views

Riemann integral with intervals?

Let $f(x) = \begin{cases} 3 && 0 \leq x \leq 1 \\ 0 && 1 \leq x \leq 2 \end{cases}$ Compute $\displaystyle \ \ \int_0^2 f(x)dx\,\,\,$. You can use the definition of Riemann integral ...
2
votes
1answer
281 views

About measure theoretic interior and boundary

Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery. I just want to clarify whether these definitions of measure theoretic interior and boundary are correct. Given ...
2
votes
1answer
226 views

dominated convergence theorem

I am studying the proof of a theorem and in a part of the proof I have the following situation: Let $u : \Omega \rightarrow R$ a nonnegative measurable function, with $\Omega$ open and bounded. ...
2
votes
1answer
421 views

Lebesgue integral is linear in simple functions.

Let $(X,\mathcal{M},\mu)$ be a measure space. Let $s,t: X\to[0,\infty)$ two simple functions. If $E\in\mathcal{M}$, show that, $$\int_E (s+t)\,d\mu=\int_E s\,d\mu+\int_E t\,d\mu$$ Attempt: By ...
2
votes
1answer
53 views

about well-defined integral kernel

Let $\phi:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ measurable function such that $$ \int_{\mathbb{R}^n}|\phi(x,y)|\ dx \leq M\ , \quad \int_{\mathbb{R}^n}|\phi(x,y)|\ dy \leq M\,.$$ Let $f\in L^p(\...
2
votes
1answer
486 views

Verifying Fatou's Lemma

Royden's Real Analysis Question: Let {$f_n$} be a sequence of nonnegative measurable functions on $R$ such that $f_n\implies f$ pointwise on $E$. Let $M\geq0$ be such that $\int_Ef_n\leq M$ for all $n$...
2
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1answer
332 views

A representation theorem for a minimally sufficient statistic by Bahadur

The Statement of the Problem I'd appreciate help in proving the following, unproven theorem from a classic article by Bahadur ([BAH], Theorem 6.3) (the expressions in square brackets are my ...
2
votes
1answer
76 views

showing to be extreme subset (might use Hahn decomposition Theorem)

I am studying Functional analysis by myself and stumbled this question and am completely at a loss. We want to show that $\{ f \in L^1 [0,1 ] : ||f|| =1 \}$ is an extreme subset of $\{ \mu \in C[0,1]^...
2
votes
1answer
63 views

Proving that $\sigma(\tau_{\mathbb{R}}\times\sigma(\tau_{\mathbb{R}}))=\sigma(\tau_{\mathbb{R}})\otimes \sigma(\tau_{\mathbb{R}})$

$\sigma(\tau_{\mathbb{R}})$ denotes the Borel $\sigma$-algebra ($\tau_{\mathbb{R}}$ is the usual topology on $\mathbb{R}$), $\sigma(\tau_{\mathbb{R}}\times\sigma(\tau_{\mathbb{R}}))$ is the $\sigma$-...
2
votes
1answer
44 views

Open bounded set $E$ so that $m(E)\neq\lim_{n\rightarrow \infty}m(O_n)$

Let $E$ be a compact set and let us define the series: $$O_n=\{x\in R^d |d(x,E)<1/n\}$$ I proved that: $$m(E)=\lim_{n\rightarrow \infty}m(O_n)$$ Now I'm trying to find an open bounded set $E$ for ...
2
votes
2answers
197 views

Part of proof 11.10 in Rudin's Principles of Mathematical Analysis

There is a part of proof 11.10 that I don't get in Rudin's Principles of Mathematical Analysis (3rd edition). The whole theorem is the following two statements: $\mathcal{M}\left(\mu\right)$ is a $...
2
votes
1answer
91 views

Why does this inequality for all characteristic functions imply it for simple functions?

This question is probably obvious, but I'm not seeing how to obtain it. A simple function is said to be finitely simple if its support is of finite measure. Let $(X_1,\mu_1)$, $(X_2,\mu_2)$, and $(Y,\...
2
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1answer
222 views

Uniformly Integrable of sets in $L_{1}(\mu)$ is equivalent to almost order boundedness

A bounded set $F\subseteq L_{1}(\mu)$ is said to be uniformly integrable if : $\forall \epsilon$ there is a $\delta>0 $, such that $\forall$ measurable set $A$, and $\forall f\in F$ , if $\mu(A)&...
2
votes
1answer
92 views

Bernstein theorem on monotone functions. Nonbounded case

Bernstein theorem on monotone functions states that bounded $C^\infty$ function $f(x) \colon (0,+\infty) \to \mathbb{R}$ satisfies inqualities $$ (-1)^n \frac{d^n f(x)}{dx^n} \geqslant 0 $$ for ...
2
votes
1answer
213 views

Measure, absolutely continuous on boundary

Let $\mu$ be a finite nonnegative Borel measure on $\mathbb R^2_+=[0,+\infty) \times [0,+\infty)$ such that $\mu( \partial \mathbb R^2_+)=0$, i.e. $\mu$ is absolutely continuous on boundary. Is it ...
2
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1answer
115 views

The measurability of the sets $\{(\omega,\omega):\space \omega\}$, $\{f=g\}$

If $\left(\Omega_0,\mathcal{A}_0\right)$ is a Borel space, the set $B:=\left\{\left(\omega,\omega\right)\space:\mid\space\omega\in\Omega_0\right\}$ is measurable in $\mathcal{A}_0\otimes\mathcal{A}_0$....
2
votes
1answer
443 views

Why does the supremum over finite partitions not suffice in defining total variation of complex measure?

In Rudin's Real and Complex Analysis, Chapter 6, eqn. 3, the total variation of a complex measure is defined as a supremum over all possible partitions of a set. Why do we need to consider all ...
2
votes
1answer
55 views

Measure of the Boundary

Let $A = [0,1/2]$U$[1/2,1]$ What is the Lebesgue measure of A? What is the measure of the boundary of A? Attempt: We can consider A to be [0,1] to obtain mA = 1. The boundary of A consists of ...
2
votes
1answer
230 views

A proof about $\sigma$-algebras via transfinite induction

This is a proofreading question. I was trying to help out on this question and in the course of that I encountered the following assertion: Let $(X, \mathcal A)$ and $(Y, \mathcal B)$ be $\...
2
votes
1answer
222 views

How to recover a measure from its Fourier transform?

Let $f$ be the complex function defined on $\mathbb{R}$ by $$ f(t)=\frac{1-it}{1+it}. $$ 1) Does there exist a complex bounded measure $\mu \in M(\mathbb{R})$ such that $\hat{\mu}=f$ (where $\hat{}$ ...
2
votes
1answer
404 views

Egoroff's theorem analogous for infinite measures

Instead of $\mu (X)<\infty$ suppose that $|f_n|\leqslant g, \forall n\in\mathbb N,$ and $g\in L^1(μ) $. $$$$ If $f_n\longrightarrow f $ a.e. in X , then prove that:$$\forall \epsilon>0, \...
2
votes
1answer
401 views

Approximating measurable function by continuous ones

Say that I have a measure space $(X,\mu)$ and a measurable function $f$ which is non-negative and bounded from above. $\mu(X)<\infty$. Now, the approximation under my concern is in almost ...
2
votes
1answer
198 views

Equivalence of measures and $L^1$ functions

Suppose we have two probability measures $\mu$ and $\delta$ on $(X, \mathcal{B})$ such that $ \delta <<\mu << \delta $. How can I prove that $f \in L^1(X,\mathcal{B}, \mu)$ iff $f \in L^1(...
2
votes
1answer
75 views

Constructing simple function for a product

Let $f,g:X\to \Bbb R_+$ be two measurable functions for some $\sigma$-algebra on $X$. Suppose that for some constant $c\geq 0$ and any $x\in X$ it holds that $f(x)g(x)\geq c$. Is it possible for any $\...
2
votes
1answer
739 views

Independence $\sigma$-algebra

I'm reading through the proof of Kolmogorovs 0-1 law and one statement is made but I can't find the proof to this statement anywhere. Could anyone help me out? The statement is given by: Let $A_1, ...
2
votes
1answer
41 views

Isomorphisme of measurable space

Hi, Can you ,help me to understand this proposition, and it's prrof ? Definition 24 is : A measurable space $(T, \mathcal{T})$ is said to be separable if there existe a sequence $(A_n)$ ...
2
votes
1answer
65 views

Small question about a lemma of measurability

Hi ; I have this lemma , and i want to ask tow questions : 1) What is the diffrence between say that $\varphi$ is measurable and to say that $\varphi$ is $(\mathcal{T},\mathcal{B}(U))$measurable . ...
2
votes
1answer
100 views

Measurable selection

I want to understand the proof of this theorem: Let $X$ be a sepable metric space, $(T,\mathcal{T})$ a measurable space, $\Gamma$ a multifunction from $T$ to complete non empty subsets > of $X$. ...
2
votes
1answer
55 views

Sigma Algebra where M is any function?

Let $M$ be any $\sigma$-field and let $m$ be a function such that $m: M \to[0, \infty]$ and for any disjoint sequence $E_n$ in $M$ $m(\cup_n E_n) = \sum_n m(E_n)$ holds. Show: 1) for $A \subset B$ ...
2
votes
1answer
131 views

Describing measurable functions for a 0-1 measure space.

This question is inspired by problem 1.6 in Rudin's "Real and Complex Analysis". There is an uncountable set $X$, and $R$ is the collection of all subsets $E$ of $X$ s.t. either $E$ or $E^C$ is at ...
2
votes
2answers
391 views

Graph of a measurable function

I am reading through Terence Tao, and I was wondering how one would prove that if $f:\mathbb{R}^d \rightarrow [0, \infty]$ is measurable, then the area under $f$ is a measurable subset of $\mathbb{R}^{...
2
votes
1answer
660 views

The outer measure of the union of bounded disjoint sets

Let $A$ and $B$ be bounded sets for which there is an $c > 0$ such that $|a - b| ≥ c$ for all $a \in A$, $b \in B$. Prove that $m^*(A \cup B) = m^*(A) + m^*(B)$. I saw this question in Royden 4th ...
2
votes
2answers
148 views

$L_p$ space,convergence

Let $1<p<\infty$ and $h\in L_p(\mathbb{R})$,that is,$\left(\displaystyle\int_{\mathbb{R}}|h|^p\right)^{1/p}<\infty$. Define a sequence $(f_n)_{n\in\mathbb{N}}$ by $f_n(x):=h(x-n)$. How to ...
2
votes
2answers
402 views

How can I prove that this simple function is Borel measurable?

How can I prove that the simple function gn that is defined below is Borel measurable? Given: let $E$ be a normed space and let $X$ be a measurable space and let $f:X \rightarrow E$ is strongly ...
2
votes
1answer
171 views

Real Analysis Qual Problem 2

This shouldn't be a hard problem, but I am stuck on it. I just need to prove the statement or come up with a counterexample. Any help will be appreciated. Let $f: [0, 1] \rightarrow [0, \infty)$ be ...
2
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1answer
194 views

Collection generating the Borel $\sigma$-algebra of the collection of compact sets (Castaing-Valadier).

I'm working on an article "Castaing-Valadier" and in Chapter 2 there is this theorem: If $X$ is a separable metric space, the Borel $\sigma$-field on $\mathcal{P}_K(X)$ (the collection of compact ...
2
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1answer
135 views

Sequence of Nonnegative Measurable Functions

While working through problems, I came upon one that I couldn't figure out and was wondering how to do. Here's the problem: Find a sequence $\{f_n\}^\infty_{n=1}$ of nonnegative measurable functions ...
2
votes
1answer
113 views

Show that a collection of Borel-subsets are Borel.

Let T be a rotation of $\mathbb{R}^{2}$ about the origin. I want to show that the collection of subsets $A \in \mathbb{R}^{2}$ such that $T(A)$ is Borel is a $\sigma$-algebra. I.e. show that $\...
2
votes
1answer
184 views

Elementary Measure Theory Problem

I'm trying to solve the following exercise but I always end up missing some vital step along the way, so any help would be much appreciated! Let $\mathbb{Q}$ be the set of all real rational numbers, ...
2
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1answer
439 views

Prove that $L^1$ is a Banach algebra with multiplication defined by convolution

To be more specific, prove that $L^1(\mathbb{R}^n)$ with multiplication defined by convolution: $$ (f\cdot g)(x)=\int_\mathbb{R^n}f(x-y)g(y)dy $$ is a Banach algebra. All the properties of Banach ...
2
votes
1answer
400 views

Measurable Partition and Ergodic Decomposition

I need some background before asking the question: Let $\mathcal{P}$ is called a measurable partition if there is a measurable set $M_0\subset M$ with full probability measure such that, restric to $...
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1answer
118 views

Conditional independence of components of an $n$-dimensional Hawkes process

Let's say we are given a $n$-dimensional Hawkes point process. To be more precise, let $N = (N_1,\dots,N_n)$ be a point process and $\mathcal{H}_t=\sigma\{N_1(s),\dots,N_n(s):0\leq s\leq t\}$ be the ...
2
votes
1answer
480 views

what is the definition of a $\mu$-measurable function?

Following was asked in a question by jpv (which is in turn pointed out by t.b. to me): Let $(X, \mathcal{F}, \mu)$ be a measure space and $(Y,d)$ be a separable metric space ($d$ is the metric). ...
2
votes
1answer
112 views

rearrangement is non-expansive

I found this statement about rearrangement from analysis Lieb and Loss in chapter 3. Suppose f, g are nonnegative functions in $L^2(\Bbb{R^n})$, then $||f^*-g^*||_2 \le||f-g||_2$ Where $f^*$ is the ...
2
votes
1answer
812 views

Lebesgue measure, Borel sets and Axiom of choice

I cannot proceed my study on measure theory since it seems my measure theory is really unstable. I desperately need someone to briefly answer below 3 questions... **For convenience, i will write ...
2
votes
1answer
90 views

How to apply product object in category to get product sigma algebra/topology/set systems?

Following is similar to my earlier questions, but try to understand them from category theory. Mariano said it was possible in a comment, but I don't know how. An object $X$ is the product of a ...