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

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1answer
67 views

Converging almost surely and value of the integral

Given a sequence of nonnegative functions $(f_n)_n$ converging almost surely (for the Lebesgue measure $d\mu$) to $f$. Assume that $\int f_n d\mu \rightarrow c < \infty$ as $n$ goes to infinity. ...
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1answer
148 views

Proving $\mu(A)=\inf\{\mu(O) \mid A\subseteq O, O \text{ open}\}$

Can someone please help me show, why in a compact metric space $(X,d)$ we have have $$ \mu(A)=\inf\{\mu(O) \mid A\subseteq O, O \text{ open}\}$$ and $$ \mu(A)=\sup\{\mu(K) \mid K\subseteq A, K \text{ ...
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2answers
217 views

Almost sure convergence of random variable

I see a lot of examples of limit theorems in terms of functions, and sequences of functions. But I think the transition from the general measure space to the probability space ...
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1answer
156 views

How to prove the limsup equals liminf for a monotone class.

How to prove if a class is monotone, then its limit supremum equals its limit infimum. Example, ${A_{n}}$ is a monotone class with $A_{n} \subset \Omega$, and $A_{1} \subset A_{2} \subset A_{3}... $, ...
5
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3answers
101 views

How to derive a union of sets as a disjoint union?

$$\bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty (A_{1}^c \cap\cdots\cap A_{n-1}^c \cap A_n)$$ The results is obvious enough, but how to prove this
4
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1answer
61 views

How to show that Vitali set can't be nowhere dense in $[0,1)$

I saw a comment mentioning that it can be shown "a Vitali set cannot be nowhere dense, nor even meager" by Baire category theorem. But I don't know how. In particular, assuming $\bf{AC}$, the ...
2
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1answer
305 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 ...
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3answers
78 views

Convergence in measure of characteristic functions

I was having trouble starting this problem. I would appreciate some help. Thanks in advance. Let $E_1, E_2, \ldots$ be measurable sets. Suppose that the functions $f_j = 1_{E_j}$ converge in ...
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0answers
62 views

Using $\frac{1}{A+i\epsilon} = PV\frac{1}{A}-i\pi\delta(A)$ in Feynman Integrals

Is the following operations OK (this is related to the Feynman parameter trick)? $$F:= \int_0^1 \mathrm{d}x\int_0^{1-x}\mathrm{d}y \frac{1}{f(x,y)+\mathrm{i}\epsilon}.$$ Now using ...
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2answers
111 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 ...
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2answers
148 views

Question from Folland on modes of convergence

I have been reading through Folland, and I am having a hard time answering the following question. Any help will be much appreciated. Suppose $\lvert f_n \rvert \leq g \in L^1$ and $f_n \rightarrow ...
4
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1answer
80 views

Double Jumps of a Poisson Process

If $N_t$ be a Poisson Process with rate $\lambda>0$, surely for any prescribed $t>0$, the probability that $N_t$ "jumps (at least) twice" at $t$ is zero, i.e. ...
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2answers
371 views

If $\{f_n\}$ is a measurable sequence of functions, then $\{x : \lim f_n(x) $ exists $\}$ is measurable

Was hoping someone could help me out on this problem (self-study not hw). If $\{f_n\}$ is a sequence of measurable functions on $X$, then $\{x : \lim f_n(x) $ exists $\}$ is a measurable function. ...
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0answers
111 views

smallest sigma algebra can be obtained by countable operations?

Let $\mathcal{A}\subset \mathcal{P}(X)$. Then $\sigma(\mathcal{A})$, the smallest $\sigma$-algebra containing $\mathcal{A}$ is given by the intersection of all $\sigma$-algebras containing ...
5
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1answer
260 views

Billingsley's solution to 3.18 (b)

Problem 3.18(b) in Billingsley's Probability and Measure (3e) is Show that if $\lambda^*(E)>0$, then $E$ contains a nonmeasurable subset. [Here $\lambda^*$ is the Lebesgue outer measure.] ...
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1answer
59 views

σ-algebra generated by class

Let $C$ denote the collection of singleton's $\{q\}$, where $q$ is rational; that is, $$C=\big\{\{q\}:q\in\Bbb Q\big\}$$ Show that the smallest $\sigma$-algebra containing $C$ is $$F = \{ A : A ...
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2answers
325 views

The space of Riemann integrable functions with $L^2$ inner product is not complete

I am trying to find a sequence of Riemann integrable functions on $[0,1]$ converging in $L^2$ to a Lebesgue but not Riemann integrable function. I tried Dirichlet function but could not find a ...
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1answer
95 views

Atomless measure algebra is direct sum of at most countably many homogeneous measure algebras

How can we prove the following theorem from Measure Theory Vol II (page 280) by Vladimir? 9.3.5. Theorem. (i) Every atomless measure algebra is the direct sum of at most countably many ...
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1answer
201 views

Is the $\sigma$-algebra generated by a product topology a subset of the product $\sigma$-algebra of the individual topologies?

Is the generated $\sigma$-algebra of a product topology a subset of the product $\sigma$-algebra of the individual topologies? Formally, let $\Theta$ be some non-empty set (to serve as a set of ...
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1answer
80 views

simple question from set theory/measure theory

This is a simple question. On pages 5-6 of Measure Theory,Vol 1, Vladimir Bogachev he writes that: for $E=(A\cap S)\cup (B\cap (X-S))$ Now, he writes that: $X-E = ((X-A)\cap S) \cup ((X-B)\cap ...
3
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2answers
126 views

Sets with measure zero are closed under countable unions

If $(X,\chi,\mu)$ is a measure space and $Z = \{ E \in \chi \mid \mu(E)=0 \}$ then $Z$ is closed under countable union. In the other parts to this question, I showed that given any $E \in \chi$ ...
3
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1answer
148 views

Rudin Real And Complex Definition 2.16

I'm having some difficulties in the remark Rudin makes in definition 2.16, when he says that , if we consider the situation described in theorem 2.14(Riesz representation theorem), if $E$ is in the ...
4
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1answer
127 views

I don't understand the definition of completion of a $\sigma$-algebra

I am preparing for the test there is a question that I dont understand: suppose that $(\Omega,\mathscr F,P)$ is the probability space where $\Omega:=\{1,2,...,6\}$, $\mathscr ...
4
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2answers
118 views

Help! I have proven that the Area of a $1\times 1$ Square is $0$

Let the square $S$ be the set of points $(x,y) \in [0,1]^2$ Let $R \subset S = S \cap \mathbb{Q}^2$, that is, the "rational pairs" in the square. To each of these points $r_i \in $ R, we can ...
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0answers
37 views

Lebesgue-Stiletjes measure on $\mathbb{R}^2$

Let $F$ be a two variable continous function that satisfy \begin{equation} F(x_1,y) \leq F(x_2,y) \text{ for } x_1\leq x_2, \end{equation} \begin{equation} F(x,y_1) \leq F(x,y_2) \text{ for } y_1\leq ...
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0answers
72 views

Is the measure evaluation map measurable?

I have a separable metric space $X$ from which I form the space of Borel probability measures on it, $\Delta(X)$. This last set is metrized by the topology of weak convergence (or by the Prokhorov ...
2
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2answers
300 views

A question on Lebesgue measurable functions

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a Lebesgue measurable function, and define a function $\phi:\mathbb{R}^2\rightarrow\mathbb{R}$ by $\phi(x,y):=f(x-y)$. I want to prove that $\phi$ is ...
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1answer
91 views

Is it true that probability measures are equal

I have two Borel probability measures $\mu_1$ and $\mu_2$ with support on $\left\{ (x,y) \neq 0 \mid x\geqslant 0, y \geqslant 0 \right\}$ and such that for any $v \geqslant0, w\geqslant 0$ we have $$ ...
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1answer
176 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
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1answer
145 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
79 views

Convergence in measure and metric notion

Any help with this problem is appreciated. Given a measurable set of finite measure $D$, define $L_0(D)$ to be the vector space of real valued measurable functions on $D$. Define $d(f,g) := \int_D ...
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0answers
65 views

Completely Regular Topological Space and Measure Theory

Here is the statement... Suppose that $(X,\tau)$ is a comletely regular topological (I think the lecturer requires X to be Hausdorff too.), and that E is a dense linear subspace of ...
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1answer
26 views

Show that measure is trivial

Let $\mu(dx,dy)$ be a Borel measure with compact support on $\left\{(x,y) \mid x>0, y>0 \right\}$ and let $\mu(dx,dy)$ satisfy $$ \mu(dx,dy) = \lambda^{\alpha} \mu \left( d \frac{x}{\lambda}, ...
3
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1answer
334 views

Real Analysis Qualifying Exam Problem

I think this should be an easy question, and I believe the answer should be in the positive, but I am not sure how to start. I would appreciate some help. Thank you. Suppose that $f_j$ is a ...
2
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1answer
1k views

What is levy measure? Why is it needed, and what is $(1\wedge|x^2|)$?

A Borel measure $\nu$ on $\mathbb{R}$ is called a Lévy measure if $\nu({0})=0$ and $\int_\mathbb{R}(1\wedge|x^2|) \, \nu(dx) < \infty .$ ...
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1answer
147 views

A classic example in measure theory, hold the measure theory

I answered a question recently drawing on the following construction. Let $\{r_n\}$ be a enumeration of the rationals and set $$E:=\bigcup_{n=1}^\infty B(r_n,2^{-n}).$$ Then $$m(E) \leq ...
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1answer
81 views

Measurable Maps

Let $X$, $M$ be two metric spaces and $\nu:X\rightarrow \mathcal{M}_1(M)$, $x\mapsto \nu_x$ a map, where $\mathcal{M}_1(M)$ is the space of all probabilities over $M$ with the Boral $\sigma$-algebra, ...
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1answer
169 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 ...
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1answer
52 views

$A\subset[0,1]$ with measure $<1$ so that $\int_A f(x) dx = \int_{[0,1]} f(x) dx$ for continuous $f$?

Is there a set $A\subset[0,1]$ of measure $<1$ so that $\int_A f(x) dx = \int_{[0,1]} f(x) dx$ for continuous $f:[0,1] \to \mathbb{R}$? $A^c$ must have empty interior for sure. (Otherwise support ...
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3answers
260 views

Is there a dense subset of [0,1] of measure 1/2 whose complement is also dense?

I want to find a set $A \subset [0,1]$ so that: $A$ is dense in $[0,1]$ $A^c$ is dense in $[0,1]$ $A$ is Lesbesgue measure $1/2$ (Failing this....I want both sets to be positive measure) My first ...
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1answer
176 views

About a proof that elements in a certain $L_2$ convergent series are also in $L_\infty$

The problem I have is about convergence of series expansions of random fields (or stochastic processes, if you will), which don't converge in the norm I want, that is $L_\infty$, but in $L_2$. I have ...
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1answer
90 views

Subadditivity of Measure, how to prove

I want to understand the following inequality, where $B, A $ and $C$ are sets in $X:$ $$ \mu(B \cap A) + \mu((B \setminus A) \cap C) + \mu((B \setminus A) \setminus C) \geq \mu(B \cap (A \cup C)) + ...
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1answer
79 views

Show that a measurable function is integrable.

Let $x$ be a nonnegative random variable on a probability space $(X, \Sigma, P)$ such that $\mathbb{E}(x)=1$ and $P\{x>0\}=1$. Define $y:X\to \overline{\mathbb{R}}_+$ by $$y(w)=\frac{1}{x(w)}$$ if ...
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1answer
34 views

Use of units of measure, from more precise to less precise. [closed]

When should you switch from using a more precise unit of measer, to using a less precise unit of measure? When refering to an elapsed period of time, when would you swich form using: seconds - ...
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1answer
106 views

Stieltjes measure

Given a nowhere dense uncoutable compact subset $K$ in $\mathbb R$, can we find a Stieltjes measure $\mu$ such that $\mu(K)=1$, $\mu(\mathbb R\setminus K)=0$ and every single point in $K$ has measure ...
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1answer
97 views

Is a product that contains a non-measurable factor measurable in the product $\sigma-$algebra?

I have the following question: Let $\mathcal{B}([0,1])$ denote the Borel $\sigma$-algebra on $[0,1]$. Suppose $A\subset [0,1]$ is not Borel-measurable. Is the set $A\times [0,1]\times ...
6
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2answers
291 views

Weakening Carathéodory's criterion?

Assume that $E$ has finite outer measure. Show that $E$ is measurable iff for each open, bounded interval $(a,b)$, $$b-a = m^*\big((a,b)\cap E\big) + m^*\big((a,b)/E\big)$$ This is an exercise on ...
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1answer
110 views

almost surely convergence to zero of indicator function

How can we show that: if $x$ is an integrable function (i.e., $\int_X|x|<\infty$), and $\{S_m\}$ is a sequence in $\Sigma$ ($\Sigma$ is a $\sigma$-algebra on $X$) in $X$ such that $\mu(S_m)\to 0$ ...
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1answer
179 views

Prove Convergence in Probability is closed under multiplication

This seems like a pretty plain question, but I can't figure it out. Let $X_n \to X$ in probability, and $Y_n \to Y$ in probability. Show that $X_N Y_N \to XY$ in probability. So far I can only show ...
2
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1answer
93 views

Probability Curiosity

Let $X_{1},X_{2},\ldots$ be i.i.d. random variables with $ E\left[X_{1}\right]=0$ and $0<Var\left(X_{1}\right)=\sigma^{2}<\infty.$. Let $ S_{n}=\sum_{j=1}^{n}X_{n}$. Consider now ...