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

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Proving that $\int_{\mathbb{R}} f \ d\mu = \frac{1}{N}\sum_{i=1}^N f(\lambda_i)$

I want to know if my proof is correct and if there is some easier way to prove this (you don't need to read all my proof, I'm accepting as answers another proofs, not just corrections of mine). ...
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1answer
25 views

From nowhere dense perfect set to zero measure set.

I know that Cantor set is nowhere dense and perfect. But if I have a nowhere dense perfect set, can I call it a Cantor set? Also, I already proved that a certain subset of the real line is a ...
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18 views

Product of measure spaces

Show that B(R^n)=B(R)*B(R)*B(R)...n times where B(R) is a Borel sigma algebra of R. I know B(R^n) subset of B(R)*B(R)*B(R).. But I couldn't get idea of reverse inclusion. Please help me out.
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13 views

Why is “having countably many open rays” a measurable condition

In discussing Bernoulli($p$) percolation on a tree, sometimes one asks the question of what the probability is that there are countably infinitely many rays containing only open edges. I don't see ...
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24 views

Understanding averaging of symplectic matrices via Haar measure

In McDuff and Salamon's Intro. to Symplectic Topology (2nd edition), there's a proof that $U(n)$ is a maximal compact subgroup of $Sp(2n)$ which I'm trying to understand. The proof uses the Haar ...
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2answers
73 views

Strange behaviors of finitely additive probabilities

Watching a lecture on youtube I heard the lecturer stating that in general finitely additive probabilities behaves strangely. For example, it is possible that every open interval around a point $x$ ...
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0answers
12 views

Transfer Lebesgue measure on $\mathbb{R}^2$ to $\mathbb{C}_{\infty}$

I've got a quick question. Let $\lambda$ denote the Lebesgue measure on $\mathbb{R}^2$. I want to express the following "intuitive statement" mathematically: Since $\mathbb{R}^2\cup\{\infty\}\simeq ...
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1answer
20 views

Limit of translates of characteristic function

This might be silly, but what is a simple way of showing that given a characteristic function of a lebesgue measurable set in $\mathbb{R}$ then we have $\lim_{t \rightarrow 0} \chi (x-t) \rightarrow ...
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30 views

Trouble understanding some basic concepts of measure theory [on hold]

I am currently undergoing a course in Measure Theory. The book is "Principles of Real Analysis" by Charalambos D. Aliprantis and Owen Burkinshaw. The approach is little difficult for me to grasp and I ...
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0answers
29 views

If $\mu(E)>0$ then $\exists E'\subseteq E$ such that $0<\mu(E')<\infty$.

Which measure $\mu$ have the property that for every measurable set $E$ with $\mu(E)>0$ there exist a measurable subset $E'\subseteq E$ such that $0<\mu(E')<\infty$? At first I thought every ...
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1answer
32 views

Separability of a sigma algebra

Let $E$ be a class of subsets of a space $X$ and $B:=\sigma(E)$ be the $\sigma$-algebra generated by $E$, i.e. the smallest $\sigma$-algebra that contains $E$. Let $x,y \in X$ be such that for all $A ...
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0answers
25 views

Computing $\pi_1(\text{Pr}(S),\mathbb{P}_0)$

Let $(S,d)$ be a complete separable metric space, and consider the space $\text{Pr}(S)$ of probability measures on $S$ that are defined on Borel sets arising from the metric $d$. Now endow ...
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2answers
47 views

Continuity of a Lebesgue indefinite integral over unbounded interval

We know that if $f : [a,b] \rightarrow \mathbb{R}$ is Lebesgue-integrable, then $$ F(x) = \int_{a}^{x} f(t) dt $$ is continuous. But if $f : \mathbb{R} \rightarrow \mathbb{R}$ is ...
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2answers
37 views

Lebesgue Integral over set of measure of zero

Is it defined for a non-measurable, non-negative function? It would make sense, as clearly $s=0$ is a simple function, and $s\leq f$ for any $f$, whether it is measurable or not. So the clasically ...
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1answer
26 views

Sigma-fields and probability

I'm unsure what this question asks of me. For (i) I have given a power set with 16 elements in terms of a,b,c and d. I don't understand what I need to do for (ii). I believe (iii) is fairly ...
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0answers
19 views

Showing $\{x\in \mathbb{R}:\mu(\{x\})>0\}$ is a countable set under certain conditions.

Let $\mu$ be a measure such that $(\mathbb{R}, \mathcal{B}_{\mathbb{R}}, \mu)$ is a $\sigma$-finite measure space. I have to prove that $D=\{x\in \mathbb{R}:\mu(\{x\})>0\}$ is a countable set. Let ...
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0answers
20 views

Dominated Convergence Theorem.

Dominated Convergence Theorem "Suppose $X_{n}\rightarrow X$ a.s., and there is a random variable $Y$ with $E[Y]<\infty$ such that $|X_{n}|<Y$ for all $n$. Then $E[lim_{n \to ...
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1answer
30 views

Measure Theory, $\sigma$-algebra Folland Problem 23

I'm preparing for my exam. Can anyone help me in this matter, is confusing to me thank you very much.
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21 views

Limit definition of Sets.

Proposition 1.32 $X_{n}\xrightarrow{a.s.} X$ if and only if for any $\epsilon>0$ $P( | X_{n}- X |<\epsilon, \; \forall n\geq m )\rightarrow1$ $as$ $ m\rightarrow\infty$ Proof. Suppose first ...
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1answer
21 views

The measurability of $f(x) = \sum_{r_n \leq x} \frac{1}{2^n}$

Let $\mathbb{Q} \cap [0,1] = \{ r_1, r_2, \ldots \}$ be an enumeration of the rationals and let $f : [0,1] \rightarrow \mathbb{R}$ defined by $$ f(x) = \sum_{r_n \leq x} \dfrac{1}{2^n} $$ I need to ...
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0answers
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Lebesgue-Stieltjes Measure associated to $F$.

I would like some help here, please. First is confusing to me the definition of: Lebesgue-Stieltjes Measure associated to $F$. I'm reading Folland-Real Analysis, page 35, second paragraph. I do ...
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1answer
35 views

Calculate $\displaystyle\lim_{n\rightarrow \infty}\displaystyle\int_{1}^{\infty}{\dfrac{\sqrt{x}\log{nx}\sin{nx}}{1+nx^{3}}}$

I have to calculate (if it exists) $\displaystyle\lim_{n\rightarrow \infty}\displaystyle\int_{1}^{\infty}{\dfrac{\sqrt{x}\log{nx}\sin{nx}}{1+nx^{3}}}$. I think I have to use Lebesgue dominated ...
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0answers
15 views

Is there always a g in a compact connected Lie group whose powers equidistributes in G?

I'm starting to understand some basics things about equidistributed sequences and i found my self asking this question on the basis of the example of the torus and Weyl equidistribution theorem: ...
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0answers
41 views

Existence of certain measure on $[0, 1]$

Does there exist a measure $\mu$ on the Borel-$\sigma$-algebra of $[0, 1]$ such that $\int f d\mu = \lim_{x \to 0} f(x)$ for every increasing $f: [0, 1] \to [0, \infty)$ I have no idea on where to ...
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2answers
21 views

Clarification from old post: Union of sigma-algebras is non sigma-algebra

I have been working on slightly different problem from one posted back in 2013 here. I followed closely the hints given by @martini there, but nevertheless I still got stuck. I am retyping the ...
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1answer
21 views

Is every measure translation invariant?

Is every measure translation invariant? I ask the question because I noticed that this desideratum is always required when one introduces the Lebesgue measure, but is not mentioned in the general ...
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0answers
13 views

Conditional expectation of derivative to short-form notation

I have a continuous random variable $V_t$ for which I was able to show that $$\mathbb{E}_t\left[\frac{dV}{dt}\right]=X_t.$$ I now want to write (in short-form notation) $dV_t=X_tdt$. How could I ...
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1answer
43 views

Probability of a nonnegative submartingale converging to zero [on hold]

Suppose that $\{X_k\}$ is a nonnegative submartingale, and $\Pr(X_1 = 0) = 0$. Then could we conclude that $\Pr(\liminf X_k=0) = 0$? What about $\Pr(\lim X_k=0) = 0$? Thanks a lot. Some background ...
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66 views

If $\int _{-\infty}^{\infty}f=1$ then prove that $\int_{-\infty}^\infty\frac{1}{1+f(x)}=\infty$

Given that $f:\mathbb R\rightarrow (0,\infty)$ is a measurable function. If $\int _{-\infty}^{\infty}f=1$ then prove that $\int_{-\infty}^\infty\dfrac{1}{1+f(x)}=\infty$ Any hints on how to proceed ...
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0answers
28 views

How to obtain a certain expression as an expectation

I have a probability space $(\Omega, M, \mathbb{P})$, where each $\omega \in \Omega$ is a random subset of natural numbers (i.e. This is a probability space of sequence of natural numbers sometimes ...
3
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2answers
51 views

Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$

Question: Let X be a nonnegative random variable and $0 < \lambda \leq EX$. Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$ At first glance I thought I could use some ...
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20 views

Uniform integrability of specific sequence of RV

I am investigating the following limit $$ \lim_{n \to \infty} E \left[ n \ln^-\left(1 - 2 \frac{\sigma}{n} [{\cal N}]_1 + \frac{\sigma^2}{n^2} \underbrace{ \| {\cal N} \|^2}_{\chi^2 \mbox{ ...
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1answer
9 views

Smallest (sub-) Sigma algebra of a null set

Given a probability space ($\Omega,\mathcal{A} ,P$) and $N \in \mathcal{A}, N \ne \emptyset$ with $P(N) = 0$ What is the smallest sub-sigma algebra of $\mathcal{A}$ containing $N$. I'm kind of ...
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25 views

Billingsley 2.5a) [on hold]

The field $\mathfrak{F}(\mathcal{D})$ generated by a class $\mathcal{D}$ of subsets of $N$ is defined as the intersection of all fields over $N$ containing $\mathcal{D}$. (a) Show that ...
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1answer
24 views

Limsup, is there an alternative definition or am I missing the spirit of the question?

Let $X$ be the positive integers Let $H$ be $\mathcal{P}(X)$ For finite $E\in H$ $v(E)$ is the number of points in $E$. Define: ...
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1answer
29 views

Lebesgue integral of a positive function on a set of positive measure

Let $E$ be a subset of $\Bbb R$ with positive Lebesgue measure, $\lambda(E)>0$. Let $f$ be a function from $\Bbb R$ to $\Bbb R$ which is positive on $E$, that is $f(x)>0$ for all $x\in E$. Is ...
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0answers
26 views

Is this an outer measure, if so can someone explain the motivation

I'm studying Measure Theory, and following Halmos's book and measures came first. I like measures, I also picked up a book on Probability, that motivated measures (well specifically probability ...
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1answer
69 views

“+”-Sets are measurable.

$A$ is a subset of $\mathbb{R}^2$ that for every $(x,y) \in A$ there is a $\delta >0$ that $(x-\delta , x+\delta) \times \{y\}$ and $\{x\} \times (y-\delta , y+\delta)$ are subsets of $A$. prove ...
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2answers
43 views

Measure theory exercise

From measure theory volume 1 by Fremlin, exercise 111Xf: Let $X$ be a set, $\mathcal{A}$ a family of subsets of $X$, and $\Sigma$ the $\sigma$-algebra of subsets of $X$ generated by $\mathcal{A}$. ...
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1answer
17 views

Easy argument in Lemma of Corea formula

I don't understand a presumably easy argument in my textbook. Let $L: \mathbb R^n \to \mathbb R^m$ be a linear map, $n \geq m$, $A \subset \mathbb R^n$ $\lambda^n$-measurable. We assume that $\dim ...
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2answers
16 views

Convergence of a function involving a characteristic function of a decreasing interval

Let $f_k: \mathbb{R} \rightarrow \mathbb{R} , f_k(x) = \frac{1}{\sqrt{x}}\chi_{\left[\frac{1}{2^{k+1}},\frac{1}{2^k}\right]}(x)$ For $k \rightarrow \infty $ the interval ...
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1answer
26 views

Algebra vs. Sigma-Algebra Condition

I just wanted to clarify the difference between the Algebra and $\sigma$-algebra: Algebra: If $A_1, A_2 \ldots $ are in $\mathscr A$, then $\bigcup_{i = 1}^{n} A_i \in \mathscr A$ ...
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1answer
25 views

Transformation theorem: calculate picture of a set

I have this function: $T:(0,\infty)^2 \rightarrow T((0, \infty)^2), \quad T(x,y)=\left( \frac{y^2}{x},\frac{x^2}{y} \right)$ Now I try to estimate $T(M)$ with: $0<p<q, \quad 0<a<b$ ...
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0answers
24 views

Differentiation through the integral sign, more general case

I wondered in which cases, given a measurable space $(A,\mu)$, Banach spaces $E,F$, an open $U\subseteq E$ and $f:A\times U\rightarrow F$, we can conclude that the function $s\mapsto\int_A f(x,y)dx$ ...
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0answers
16 views

Two random vectors converge does this mean that the entries converge?

Suppose you are given the following two equalities $\mathbf{\delta }^{n}=\left( \begin{array}{ccccccc} \delta _{n,1} & \delta _{n,2} & \cdots & \delta _{n,n} & 1 & 1 & ...
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1answer
41 views

Showing $\sum_{n\in\mathbb{N}}\int{|f_{n}-f|d\mu}<\infty$ implies $f_{n}\rightarrow f$ almost everywhere.

Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of integrable functions and $f$ an integrable function. I have to show that, if $$ \sum_{n\in\mathbb{N}}\int{|f_{n}-f|d\mu}<\infty, $$ then ...
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0answers
39 views

Multiplication rule and regular conditional probability

I've been studying the conditions of existence of the regular conditional probability and have a question about it. Let's $(\Omega, \mathcal{B}, P)$ be a product probability space, and let's say the ...
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0answers
20 views

estimating a convolution type maximal function

Let $\phi : \mathbb{R}^n \rightarrow \mathbb{R}_{+}$ be a $C^1$ function with $supp(\phi) \subset B(0,1)$ and $\int \phi = 1$. Define $$\phi_t(x) := t^{-n} \phi({x/t})$$ and set $$ M_{\phi} f(x) := ...
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1answer
17 views

Integration with respect to a measure

I am trying to get an explanation in words, or math, of what the $d\mu$ means in an integration statement. Such as: $$\int f \ d\mu$$ How does the measure change our old "calculus" notion of ...
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1answer
24 views

Showing that $\mathcal{A}$ being countable $\Rightarrow f(\mathcal{A})$ is countable - (Algebras/Sigma Algebras)

For the first question my idea was to show that $\sigma(f(\mathcal{A})) \subseteq \sigma(\mathcal{A})$ and $\sigma(\mathcal{A}) \subseteq \sigma(f(\mathcal{A}))$. As for the second question I am at ...