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

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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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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 ...
3
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0answers
15 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
69 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
11 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
13 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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0answers
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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27 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 ...
2
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0answers
24 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
38 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
25 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
19 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
20 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
22 views

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
12 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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0answers
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 ...
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2answers
50 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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0answers
19 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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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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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
40 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
38 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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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
22 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 ...
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1answer
36 views

Showing that if $A_{1},A_{2},…$ are all algebras then the union of all of them is an algebra [duplicate]

I am not sure how to show this. It seems obvious but maybe its not. The help would be greatly appreciated!
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1answer
25 views

Complex Measures: Absolute Continuity [on hold]

Note: This is a lemma for: Spectral Measures: Riemann-Lebesgue Given a positive measure: $$\lambda:\mathcal{A}\to[0,\infty]$$ Consider a complex measure: $$\mu:\mathcal{A}\to\mathbb{C}$$ How to ...