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

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0
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1answer
32 views

Prove that $f$ is a continuous function at $x_0$ if and only if $\mu(\{x_0\})=0$.

Let $\mu$ be a finite measure on the $\sigma$-algebra of the Borel set in $\mathbb R$. For every $x\in \mathbb R$, $f(x)=\mu(-\infty ,x)$. Please help me prove that $f$ is a continuous function at the ...
1
vote
1answer
27 views

Fatou, Dominated Convergence, etc. for nets (in relation to stochastic processes)

In textbooks on Stochastic Processes, they always seem to assume that Fatou and DCT etc. can be applied to continuous-time stochastic processes $(X_{t})_{t\in\mathbb{R}_{+}}$. But in every book on ...
1
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0answers
48 views

Relation between a.s. and L_{2} convergence

I'm working through a proof, where I need to establish that $X_{t}\overset{a.s.}{\longrightarrow}0$. All I know is that $\left|X_{t}\right|\leq\left|Y_{n}\right|+\left|Z_{n}\right|$ for $t\in[n,n+1)$, ...
2
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1answer
46 views

integration with respect to a sequence of measures [closed]

can someone help me? Let $(\mu_n)$ be a sequence of measures on $(X, \mathcal{A})$ and $\mu:= \sum_{n \geq 1} \mu_n$. Then, for every measurable $f: X \rightarrow [0,\infty]$, $$\int_X f d\mu = ...
2
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1answer
45 views

Measure of the reciprocal of a Cantor set

I have recently started studying measure theory and as is usual we started out by calculating the measure of the Cantor set. Now I had this question in my mind as to whether the set generated by ...
1
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1answer
20 views

Question about Haar Measure from Halmos

Halmos (Measure Theory, 1950, p. 256) poses the question: Given a Locally Compact Group $G$, compact subsets of measure zero, $C$ and $D$, is the group product, $P=CD$, which is also compact, also of ...
4
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4answers
85 views

Explanation Borel set [closed]

I'm getting totally crazy with the Borel set, I use this concept for maybe two years, but I still don't know what it really is. I would really appreciate if somebody could finally explain me what is a ...
1
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1answer
28 views

How to show that if $f$ measurable, therefore $g$ define by $g(x,y)=f(x)$ is also measurable?

Let $\mu$ a measure on a space $X$ s.t. $\mu(X)<\infty $ and let $f:X\to\mathbb R$ measurable. Show that $g:X\times X\to\mathbb R$ define by $g(x,y)=f(x)$ is measurable. I know that $$\{x\in ...
1
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1answer
16 views

How to show that a integrable function is finite a.e.?

I want to show that an integrable function is finite a.e. Then I have to show that if $$\int fd\mu<\infty \implies \mu(\{x\mid |f(x)|=\infty \})=0$$ for a measure $\mu$. My idea is: $$\{x\mid ...
0
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0answers
14 views

Why is $l(t,x,\omega)=\lim_{\varepsilon\downarrow 0}\frac{1}{2\varepsilon}\int_{0}^t1_{[x-\varepsilon,x+\varepsilon]}(X_s(\omega))ds$

Currently I am reading the book "Brownian motion and stochastic flow systems" (Harrison) and in chapter 1 paragraph 3 he states the following deep theorem about Brownian motion: Theorem Let ...
1
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3answers
28 views

Area of set-difference

Let $X$ and $Y$ be two open sets in $\mathbb{R}^2$, with $X\subsetneq Y$. Is it possible that $\text{Area}(Y\setminus X)=0$? Is it possible that $\text{Area}(Y\setminus Closure[X])=0$?
0
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1answer
26 views

If $\mathbf{X}$ is a sigma-algebra on $X$ and $A, B \in \mathbf{X}$, then $A-B \in \mathbf{X}$.

I know a sigma-algebra must satisfy three conditions: contain the empty set and the whole set, be closed for complements, and be closed for countable unions (and therefore also closed for countable ...
1
vote
1answer
21 views

Outer measure induced by measure, equality of subsets

Let $(X,\mathcal{M},\mu)$ be a measure space such that $\mu(X)=1$, and let $\mu^{*}$ be the outer measure induced by $\mu$. Suppose $E\subset X$ satisfies $\mu^{*}(E)=1$. If $A,B\in \mathcal{M}$ and ...
0
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0answers
57 views

Show that the Lebesgue measure of a solution set is finite

I would really appreciate some feedback on the following proof. I think/hope there might be a "less wordy" solution (using basic real analysis/measure theory), if so, any hints would be ...
0
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0answers
27 views

Is my proof of a dominated convergence corollary using Egorov's theorem right?

Theorem: If $\mu(\Omega) < \infty$ and the $f_n$ are uniformly bounded, then $f_n \to f$ almost everywhere implies $\int f_n d\mu \to \int fd\mu$. This is a simple consequence of Lebesgue ...
1
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0answers
24 views

Define Radon measure as an integral

Let $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ be an outer Radon measure and $f \in L^1_{loc}(\mathbb R^n, \mu)$, $f \geq 0$ on $\mathbb R^n$. Now we define an outer measure $\nu: \mathbb R^n \to ...
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2answers
40 views

Convergence on every compact set implies convergence almost everywhere

Suppose I have a sequence of functions {$u_n$} that converges to $v$ uniformly on every compact subset of $\mathbb{R}^n$. Suppose further that {$u_n$} converges to $u$ in $L^1$ for every compact ...
0
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1answer
34 views

How can I evaluate this integral? - measure theory

Let $u \in L^{p}(\Omega)$, where $\Omega$ is an open bounded domain in $R^n$ and $2\leq p <\infty$. Let $u_{+}$ the positive part of $u$. I am trying to show that to show that $\int_{ \{ u \leq 0 ...
2
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1answer
31 views

If $f \in L^1 \cap L^2$ is $L^2$-differentiable, then $Df \in L^1 \cap L^2$

Working with the definition that $f \in L^2(\mathbb{R})$ is $L^2$-differentiable with $L^2$-derivative $Df$ if $$ \frac{\|\tau_hf-f-hDf\|_2}{h} \to 0 \text{ as } h \to 0 $$ (where $\tau_h(x) = ...
2
votes
0answers
24 views

Measure of intersection of three sets

Suppose $S_{1}$, $S_{2}$ & $S_{3}$ are measurable subsets of $[0,1]$, each of measure $\dfrac{3}{4}$ such that the measure of $S_{1}\cup S_{2}\cup S_{3}$ is $1$. Then the measure of $S_{1}\cap ...
0
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1answer
48 views

Showing that $f$ is not Absolutely continuous

Frist:- I am not sure about what title this question should be. Suppose the function $f:[0,\frac{1}{2}]\rightarrow \mathbb{R}$ defined by $$ f(x) = \begin{cases} 0, & \text{if }x=0 \\ x ...
2
votes
2answers
39 views

Convergene of a sequence if and only if limsup and liminf agree

This is in terms of sets (within the context of probability measure theory) if that matters. The book says "this is equivalent" without proof, I want to prove the gap! Indicator function of a set $A$ ...
1
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0answers
32 views

Question on Proof: when two measures agree on $\pi$-system, then they agree on generated $\sigma$-algbra

Let $\mu_1, \mu_2$ be measures on $\sigma(\mathcal P)$, where $\mathcal P$ is a $\pi$-system, and suppose they are $\sigma$-finite on $\mathcal P$. If $\mu_1$ and $\mu_2$ agree on $\mathcal P$, then ...
1
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1answer
21 views

Measurability of an a.e. pointwise limit of measurable functions.

Suppose that $(f_n)_n$ is a sequence of measurable functions on a set $E$ and that $f_n \to f$ a.e.on $E$. Does this imply that $f$ is measurable? I know that pointwise limit of measurable function ...
0
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1answer
24 views

Computing a $\sigma$-algebra and conditional expectation

Consider the probability space $(\Omega_3,\mathcal{F}_3,\mathbb{P})$ where the outcome space $\Omega_3$ is all sequences of three coin tosses, the $\sigma$-algebra $\mathcal{F}_3$ is all subsets of ...
0
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0answers
13 views

Uniform Distibution Over Middle-Third Cantor Set (and Its Approximates)

Above is my question. I have done part 1, but I am unsure how to do the next part(s). Part 2 at least doesn't look that difficult, but I can't get the specifics. My main issue is that I can't get a ...
0
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1answer
16 views

distance-measure method to measure the distance between two matrixes(probability distribution)

I should find a suitable distance-measure method to measure the distance between two matrixes. The elements of such matrix is 0 to 1, and the sum of the all element is 1, so I think I could treat it ...
1
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1answer
39 views

Study the convergence of the sequence of functions $f_n(x)= \frac{f(x)}{1+\frac{|f(x)|}{n}}$ (convergence in measure, pointwise and in $ L^2(R ^d)$

Study the convergence of the sequence of functions $$f_n(x)= \frac{f(x)}{1+\frac{|f(x)|}{n}}$$ (convergence in measure, pointwise and in $ L^2(\mathbb{R} ^d)$). Let f be a measurable function such ...
2
votes
1answer
35 views

a.e. convergence of a piecewise constant function $f_h(t)=\left\lfloor \frac{t}{h} \right\rfloor \cdot h$

Let $$f_h(t)=\left\lfloor \frac{t}{h} \right\rfloor \cdot h \qquad \text{ and } \qquad f(t) =t$$ Is it true that $$f_h \stackrel{h \to 0}{\longrightarrow} f \ \text{ a.e. ? }$$ Any help will be ...
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0answers
20 views

Why do we construct the Lebesgue measure with finite measure sets before sets of arbitrary measure? [duplicate]

On page 20 of the following lecture notes, Stage 5 constructs the Lebesgue measure on finite sets before constructing it on arbitrary sets as in Stage 6: ...
2
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0answers
35 views

Completeness of $ L^{p} $ spaces and “rapidly Cauchy” sequences

http://math.harvard.edu/~ctm/home/text/books/royden-fitzpatrick/royden-fitzpatrick.pdf In the book of Royden, the completeness of $ L^{p} $ spaces has been done using what he calls "rapidly ...
4
votes
0answers
34 views

Co-Area formula in Riemannian geometry

I wonder if the following holds true: Let $z:[0,1]\times B^{n-1}_r(0)\to(M^n,g), (t,p)\mapsto z(t,p)$ be a diffeomorphism a.e. onto its image with respect to the Lebesgue measure on $A:=[0,1]\times ...
1
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0answers
22 views

Does measurability of composition and one of the functions imply measurability of the other

Consider $f:(X,\mathcal{X})\rightarrow(Y,\mathcal{Y})$ and $g:(Y,\mathcal{Y})\rightarrow\left(Z,\mathcal{Z}\right)$ . If we know that $f$ is $\mathcal{Y}-\mathcal{Z}$ measurable and $g\circ f$ ...
1
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1answer
28 views

Differentiation theorem for Radon measures

I have trouble to understand a detail in the proof of the following Theorem: Theorem: Let $\nu, \mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ be outer Radon measures, such that $\nu \ll \mu$. Then ...
1
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0answers
38 views

Why do we usually construct the Lebesgue Measure on finite outer measure sets before arbitrary measurable sets?

Question: In the typical construction of the Lebesgue Measure in 6 stages (eg. in Lebesgue Integration on Euclidean Space by Jones), authors often construct the measure on sets that have finite outer ...
0
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0answers
37 views

Cumulative Distribution Functions of random variable is a random variable

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space, $ X: \Omega \to \mathbb R \ $ a random variable. meaning:$ \quad \forall B \subseteq \mathbb R \ $ borel. $\ X^{-1}(B)\in \mathscr F$ . ...
2
votes
1answer
63 views

Violation of Fatou's theorem?

I have an assignment where the answer I have come up with violates Fatou's lemma. Clearly, I have either reached the wrong answer or misunderstood Fatou's lemma. Please help me find out which one it ...
0
votes
2answers
30 views

Showing that intersection is not empty in probability space

Given a probability measure space $(X,A ,\mu)$ where we have a finite sequence of sets $A_{1},A_{2},A_{3}\text{...},A_{n}$ that belong to $A$ such that $\sum^{n}_{k=1}\mu (A_{k})>n-1$, I would like ...
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0answers
18 views

Proving that $\dim_M{K_{\lambda,\gamma}}=\dim_HK_{\lambda,\gamma}$

Define $K_{\lambda,\gamma}$ to be the attractor of the IFS $$\bigg\{\bigg(\matrix {\lambda&0\\0&\gamma}\bigg)\bigg(\array{x\\y}\bigg),\bigg(\matrix ...
1
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1answer
39 views

Show that the class of countable unions and intersections of the elements of an algebra is an algebra.

I have the following problem: Let $\mathcal{A}$ be an algebra of subsets of a set $X$. We form the class $\mathcal{B}$ of countable unions and intersections of elements of $\mathcal{A}$, that is ...
1
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2answers
17 views

Corresponding convergence types of pointwise and uniform convergence of calculus in measure theory

What are the related convergence types of pointwise and uniform convergence definitions of calculus in measure theory? As we know from calculus, uniform convergence is $stronger$ than pointwise ...
2
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0answers
32 views

Show that $d+1$-dimensional Lebesgue measure of set $G$ equals $0$

Let $D \subset \mathbb{R}^d$ and let $f:D \rightarrow \mathbb{R} $ be measurable function. Let $G=\{(x_1,x_2,\ldots,x_d,f(x_1,x_2,\ldots,x_d))\in \mathbb{R}^{d+1}:(x_1,x_2,\ldots,x_d)\in D \} $ be the ...
6
votes
1answer
86 views

“Visualizing” Mathematical Objects - Tips & Tricks

It has been a while since I am kind of stuck with my skills concerning the visualization of mathematical objects. Here there is the problem. First of all, let me point out that I am completely ...
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2answers
39 views

A property of Radon Measures

Let $\mu: \mathcal P(\mathbb R^n) \to [0, \infty]$ be an outer Radon-measure. This means that every Borel-set $B \subset \mathbb R^n$ is $\mu$-measurable, $\mu$ is Borel-regular, i.e. for every set $A ...
0
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1answer
44 views

Function in $\mathcal{L}^p(\mu)$ for all $1\leq p \leq 2$

Let (X, $\mathscr{A}$, $\mu$) be a measure space and f $\in$ $\mathcal{L}^1(\mu)\cap \mathcal{L}^2(\mu)$. Can I ask how to show that $f \in \mathcal{L}^p(\mu)$ for all $1\leq p\leq 2$?
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0answers
27 views

Proving measurability of a function only by checking generating sets

Theorem: Suppose that $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are measurable spaces and $\mathcal{B}$ = $\sigma(\mathcal{G})$ is generated by a family $\mathcal{G}\subset\mathcal{P}(Y)$. Then $f : X ...
2
votes
1answer
28 views

Joint measure from two Markov kernels

Let $(X,\mathcal{X})$ and $(Y,\mathcal{Y})$ be two measurable spaces (Polish if you wish), let $K$ be a Markov kernel from $(X,\mathcal{X})$ to $(Y,\mathcal{Y})$ and let $K'$ be a Markov kernel from ...
1
vote
0answers
45 views

Difference between convergence in measure and almost everywhere convergence

We say$$f_n \rightarrow f$$ almost everywhere on $\Omega$ $iff$ there exist N in sigma algebra $F$ such that $\mu$(N) = $0$ and $$f_n(\omega)\rightarrow f(\omega)$$ for all $\omega$ in $N^c$ and ...
1
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0answers
39 views

the Lebesgue-Stieltjes measure induced by the Cantor-Lebesgue-Vitali function

Considering the extension on the whole $\mathbb{R}$ of the Cantor-Lebesgue-Vitali function (obtained defining $f(x)=0 \quad \forall x \leq 0$ and $f(x)=1\quad \forall x\geq 1$), I have to prove that ...
2
votes
0answers
42 views

Riemann and Lebesgue improper integral Proof

I've been trying to find some notes on the following statement: Let $f:(a,b] \to \mathbb{R}$, $f\geq 0$, and $f\in\mathcal{R}[a+\epsilon , b]$ for any $\epsilon>0$. Then $\int_a^bf=\lim_{\epsilon ...