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

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Continuity of $\mu \mapsto \mu(E)$ for $\mu$ probability measure and $E$ Borel subset

Let $X$ be a topological space endowed with the Borel sigma-algebra, let $\mathcal{P}(X)$ be the set of all Borel probability measures on $X$, endowed with the weak* topology. Fix $E$ Borel subset of ...
4
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1answer
497 views

Why is unique ergodicity important or interesting?

I have a very simple motivational question: why do we care if a measure-preserving transformation is uniquely ergodic or not? I can appreciate that being ergodic means that a system can't really be ...
-4
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0answers
31 views

Prove that the following is a field.

http://imgur.com/xMsuhBf 2.4a Honestly, I'm really confused with this notation and what a field is etc. I get the 3 conditions for it to be a field I'm just really bad at proving them. So far I ...
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1answer
19 views

Set of discontinuities of one function is smaller than that of another

Let's say I know that the set of discontinuities of a function $f$, denoted by $D_f$, has measure zero (although I don't believe that fact matters). I think that it follows that the set of ...
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0answers
10 views

Dual formulation of weak $L^p$

Let $1<p\leq \infty$. Then we have$$||f||_{L^{p, \infty}(X,d\mu)} \sim_psup\{\mu(E)^\frac{-1}{p'}|\int_E f d\mu|:0<\mu(E)<\infty\}$$ Where$||f||_{L^{p, \infty}(X,d\mu)} = ...
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0answers
22 views

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). ...
2
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2answers
186 views

Construction of the completion of a measure space

Let $(X,M, \mu)$ be a measure space. Let $\overline{M}$ be collection of $E \cup Z$ such that $E \in M$ and $Z \subset F$, where $F \in M,$ and $\mu(F) = 0.$ We also know $\bar{\mu}(E) = \mu(E).$ a) ...
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1answer
26 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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22 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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16 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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1answer
33 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

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
74 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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2answers
78 views

prove the existence of a measure $\mu$

Suppose $X$ and $Y$ are compact metric spaces and $F : X \rightarrow Y$ is a continuous map from $X$ onto $Y$. If $\nu$ is a finite measure on the Borel sets of $Y$, prove that there exists a measure ...
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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
76 views
+50

Convolution of two indicator functions can't be constant

Let $A,B \subset S^1$ be measurable sets (considering $S^1$ with say the lebesgue measure). I'm trying to prove that if the convolution $1_A*1_B$ is constant then one of $A$ or $B$ is a full measure ...
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0answers
30 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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0answers
30 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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2answers
783 views

$f$ measurable with $f=g$ a.e. then $g$ measurable

How do I prove this proposition from Royden's Real Analysis: If $\mu$ is a complete measure and $f$ is a measurable function, then $f=g$ almost everywhere implies $g$ is measurable. In proving ...
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1answer
43 views

Probability space defined by function from $X$ to $[0,1]$

Let $X$ be a non-empty countable set. If there is a function $f:X\rightarrow [0,1]$ such that $p(S)=\sum_{x\in S} f(x)$ for all $S \in 2^X$, then prove that $(X,2^X, p)$ is a probability space. My ...
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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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1answer
73 views

Is the set of all Lebesgue-measurable sets measurable?

I consider the set $X^p=\{A\subseteq[0,1]|A$ Lebesgue-measurable and $\lambda(A)=p\}$ for a $p\in (0,1]$. My objective is to construct a random variable with values in $X^p$. Therefore I need to know ...
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2answers
50 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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0answers
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
27 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
21 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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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.
3
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1answer
39 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 ...
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1answer
47 views

Measure Theory - working with unusual measures and set functions

Let $m$ define the Lebesgue measure. Let $\mu$ define the measure $\mu(A)=m(A\cap(0,1))$ for a Borel set $A$. Let $K=\bigcap \{A:A$ is closed, $\mu(A)=1\}$, $D=\bigcap \{G:G$ is open, ...
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1answer
134 views

Is a regular Borel measure on a locally compact space necessarily $\sigma$-finite?

I am trying to compile a proof of the uniqueness of Haar measure. Usually this is done by multiple-integral mumbo-jumbo, abusing left and right invariance of two potential measures and invoking ...
1
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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
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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1answer
279 views

Besicovitch Covering Lemma

We just finished our unit on covering lemma's in my analysis class and my professor proved both the Vitali and Besicovitch covering lemma's (for finite and infinite coverings) using balls. He ...
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0answers
16 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
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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1answer
40 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 ...
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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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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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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 ...
3
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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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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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30 views

Generic rank of tensors

Let the tensor product of the type $$ \underset{k=1} { \overset{m} \bigotimes } v_k$$ denote a simple tensor. As underlying fields, take $$ \underset{k=1} { \overset{m} \bigotimes } ...
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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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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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2answers
978 views

Is there a solution manual for Royden fourth edition?

I bought the fourth edition of Royden Real Analysis, this book is awesome and is quite different of third edition that has less excersices. I have the solution manual for the third edition. Is there ...