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

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38 views

Sum of independent discrete random variable

Here is my attempt of deriving the sum of independent random variable in the discrete case : $\underline{\textbf{Sum of independent random variables}}$ Let $\mathcal{C_1}, \mathcal{C_2}$ be ...
2
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2answers
139 views

Finding a “big” bounded subset of a subset of $\mathbb R$

I'm working in the following exercise. First, we define the outer Lebesgue measure, $m(A)=\inf\{\sum_n l(I_n): (I_n)\text{ is a sequence of open intervals and} A \subset \bigcup_nI_n \}$, where $l$ ...
2
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0answers
34 views

Projection measures and integrability

Let $(M, \mathcal{A}, \mu)$ a probability space, $Y$ compact metric space. Consider $\mathcal{M}(\mu)$ be the space of probability measures $\eta$ on $M\times Y$ such that $\pi_{*}\eta=\mu $ where ...
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0answers
20 views

Is the standard product measure on $[0,1]^\kappa$ regular when $\kappa$ is uncountable cardinal?

Is the standard product measure $m$ on $[0,1]^\kappa$ is regular when $\kappa$ is uncountable cardinal ?
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1answer
17 views

Reference on Riesz representation theorem for $L^p(0,T,X)$ spaces.

Brezis Functional Analysis book proves the following Riesz representation theorems for usual $L^p(\Omega)$ spaces: In what book can we find an analogous of these theorems for $L^p(0,T,X)$ spaces? ...
1
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1answer
70 views

Some properties of points with Lebesgue density equal to $1$.

I am studying Evans-Gariepy book and in corollary 1 of section 3.1.2, he prove that if $f:\mathbb{R}^N\to\mathbb{R}^M$ is locally Lipschitz and $$Z=\{x:\ f(x)=0\},$$ then $Df(x)=0$ a.e. $x\in Z$. He ...
6
votes
1answer
131 views

Separability of a set with norm $\thickapprox$ $L^1$ +$L^{\infty}$

Let $(M, \mathcal{A}, \mu)$ a complete separable probability space. Recall that complete means that any subset of a measurable set with zero measure is measure (and has zero measure) and separable ...
4
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1answer
90 views

Almost all subgroups of a Lie group are free

I am currently reading this paper by Epstein. I need help with understanding the proof. Specifically, I have the following two questions. Let $w\colon G\to H$ be an analytic mapping between ...
2
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2answers
55 views

Is a $L^1$-function which is linear near the origin in $L^p$?

Suppose you have a function $f$ on $\mathbb{R}$, such that $$\int_{-\infty}^{\infty} | f(x) | \, \mathrm{d} x < \infty$$ and $$\int_{-u}^u |f(x)| \, \mathrm{d} x = \mathcal{O}(u)$$ for $u \to 0$. ...
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0answers
16 views

Measures in differential form

I came across a question recently where I needed to verify that a given measure (in 'differential' form i.e., given as $d \mu =$ some function of $(x_1, \dotsc x_n)$ times $ \prod_{i \leq n} dx_i$) is ...
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1answer
27 views

Ergodic (equivalent characterization)

Let $(\Omega,\mathcal{B},\mu,T)$ be a measuretheoretical dynamical system. Then this system is called ergodic if $$ B\in\mathcal{B}, T^{-1}(B)=B\implies \mu(B)=0\text{ or }\mu(B^C)=0. $$ ...
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1answer
45 views

Where is “countability” used in this proposition about product $\sigma$-algebra?

The following is a proposition about product sigma algebra from Folland's Real Analysis: Proposition. If $A$ is countable, then $\otimes_{\alpha\in A}M_{\alpha}$ is is the $\sigma$-algebra ...
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0answers
31 views

Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism

Let $G$ be a compact abelian metrizable group (where the group operation is written as $+$) and $\mu$ is the Haar measure on $G$. Suppose we have a measurable function $f: G \rightarrow ...
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2answers
30 views

$ \Bbb B( \Bbb R^{n+m} ) = \Bbb B( \Bbb R^{n} ) \times \Bbb B( \Bbb R^{m} ) $

Let $ \Bbb B( \Bbb R^{n} ) $ denote a Borel algebra on $ \Bbb R^n $. Why is it true, that: $ \Bbb B( \Bbb R^{n+m} ) = \Bbb B( \Bbb R^{n} ) \times \Bbb B( \Bbb R^{m} ) $ I think, that "$ \supset$" ...
1
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1answer
44 views

Series converges but term by term integration not valid?

Give an example of a series $\sum g_n$ of Lebesgue integrable functions on $\mathbb{R}$ that converges but for which term by term integration is not valid. This is last minute exam revision so I do ...
2
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1answer
29 views

Countable generation of $\sigma$-algebras

I feel that a positive answer to the following question would be helpful in solving some exercises in introductory measure theory: Suppose $\cal A$ is a collection of subsets of a set $X$ and let ...
3
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1answer
81 views

Questions on Fubini's Theorem and $\sigma$-finite measure?

I asked a question about this a several days ago, but I think I have a better formulated question now. The reason I did not just edit the last question about this is that I feel the answers I got ...
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1answer
23 views

What is the integral of $\int_{\mathbb{N}} s d\mu$ where $\mu$ is the counting measure on $\mathbb{P}(\mathbb{N})$?

What is the integral of $\int_{\mathbb{N}} s d\mu$ where $\mu$ is the counting measure on $\mathbb{P}(\mathbb{N})$? What does it mean for $s$ to be integrable? 1. This is last minute exam revision. ...
2
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1answer
53 views

Do we need $\mu, \nu$ to be $\sigma$-finite to show $\int fg \ d(\mu \otimes \nu) = \int f \ d\mu \int g \ d\nu$?

The problem statement: Let $(X, \mathcal F, \mu), (Y, \mathcal G, \nu)$ be $\sigma$-finite and $f \in \mathcal L^1 (\mu), g \in \mathcal L^1 (\nu)$. Show that $fg \in \mathcal L^1 (\mu \otimes ...
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1answer
28 views

Sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$?

I've been trying to think of a simple example of a sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$. Suppose $f$ is the zero function for the sake of ...
1
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0answers
45 views

Measurable set of points where a measurable sequence fails to converge

Let $\{f_n\}$ be a sequence of measurable functions. Prove that the set of points $x$ such that $\{f_n(x)\}$ fails to converge as $n\to\infty$ is measurable. My first attempt was Suffices to show ...
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0answers
20 views

Uniform continuity of weighted Sobolev functions.

I am trying to show an embedding result for a weighted Sobolev space and have come to the following problem: I have a function $f: (0,a] \rightarrow \mathbb{R} $ such that: $f$ is bounded and ...
2
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0answers
18 views

pseudo inverse of a finite-to-one continuous map and measurability

Given that $\pi: X \to Y$ is a continuous onto map between compact metric spaces such that the fiber $\pi^{-1}(y)$ is a finite subset of $X$ for all $y$, is the map $y \mapsto \pi^{-1}(y)$ guaranteed ...
2
votes
1answer
58 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
51 views

How to understand uniform integrability?

From the definition to uniform integrability, I could not understand why "uniform" is used as qualifier. Can someone please enlighten me?
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1answer
24 views

The general form of a measurable set in a product measurable space

Hi everyone: Suppose that $(X,\mathfrak{M},\mu)$ and $(Y,\mathfrak{N},\nu)$ are two measure spaces and consider the product measure space $(X\times ...
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0answers
49 views

Intuition behind $\sigma$-algebras: why union?

I first came across $\sigma$-algebras in measure theory. I understand that the issue is that we want to define the sets we can give a measure to, and that if we can give a set a measure, then its ...
1
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0answers
47 views

When does an integral operator belong to the Schatten - von Neumann class in terms of its kernel?

It is well known that an integral operator $X: L^2(\mu)\to L^2(\nu)$ with kernel $k(x, y)$ belongs to the Schatten -- von Neumann class $\mathfrak S_2$ if and only if $\int |k(x, y)|^2\, d\mu(x)\, ...
4
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1answer
81 views

Proof that $(L^1)\neq(L^\infty)^\ast$

I have seen a "proof" that $L^1\neq(L^\infty)^\ast$ which goes as follows: show that there is an element of $(L^\infty)^\ast$ which is not in the image of the canonical map ...
0
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1answer
42 views

An issue with $\infty \cdot 0$ in showing that Cartesian product of a set with a null set has measure zero

Here is the problem: Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be $\sigma$-finite measure spaces. Furthermore $A\in \mathcal A$ and $N\in \mathcal B$ such that $\nu(N)=0$. Let ...
1
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1answer
54 views

Why define the Lebesgue-Integral just for measurable functions?

Usually, the Lebesgue integral, for example on Wikipedia, is defined for non-negative measureable functions as $$ \int_E f \, d\mu := \sup\left\{ \int_E s \, d\mu : 0 \le s \le f, s \text{ simple } ...
2
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1answer
70 views

Conditional probability explained?

Let $F_A$ be the CDF to the random variable $A$ ( and $B$ another independet rv), how do we get that $P(A+B \le s) = \int_{\mathbb{R}} P(A+B \le s\mid A=x ) \, dF_A(x)$ (This is probably a ...
2
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1answer
46 views

$f_n(x) = x(1-x)^n$ converges uniformly to $0$ on $[0, 1]$?

I am trying to show that the sequence of functions $f_n(x) = x(1-x)^n$ converges uniformly to $0$ on $[0, 1]$. Well at $0$ and $1$, $f_n(x) = 0$ for all $n$. So let $x \in (0, 1)$. $f_n(x)$ ...
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2answers
32 views

Jordan Content of the set $\{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, … \}$?

I have the following definition of the Jordan Content of set $E$ - inf $\{ \sum_{i=1}^n |I_i| : n \in \mathbb{N}, I_1, I_2, ..., I_n$ intervals such that $E \subseteq \bigcup_{i=1}^n\}$ That seems ...
1
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1answer
17 views

Coin tossing, two heads always followed by two tails - lim sup necessary?

In Bernoulli Space $\Omega$, let $E_n$ be the event that the $n$th toss is heads. Write down a formula in terms of the $E_n$ for the following event: “Every time two Heads appear in succession, the ...
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3answers
33 views

How many algebras of subsets of $X$ contain exactly four elements?

Let X be a set with five elements. How many algebras of subsets of X contain exactly four subsets? Well $\emptyset, X$ must be in any algebra of subsets of $X$ so that means we have to find two more ...
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1answer
11 views

Outer Measure is not Finite Additive

A cube $Q$ in $\mathbb{R}^d$ is a subset $[a_1,b_1]\times \cdots \times [a_d,b_d]$ of $\mathbb{R}^d$, where $a_i,b_i\in\mathbb{R}$ and $b_i-a_i=b_j-a_j$ for all $i,j$. By volume of cube $Q$, denoted ...
0
votes
1answer
50 views

If two measures are equal, are the integrals with respect to these measures equal?

If $\mu$ and $\nu$ are probability measures such that $\mu=\nu$, then is it true that for all measurable function $f$ $$\int fd\mu=\int fd\nu \ \ \ ?$$ It is true for integrable functions but if $ ...
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0answers
31 views

Invariant Measure

Let $\dot{x}=u(x)$ a dynamical system ($x\in\Gamma$) with solution $x(t)=\Phi^t_u(y)$ and $\mu$ a $\Phi^t_u$-invariant measure on $\sigma_\Gamma$. I want to show that the smooth density $\rho=d\mu/dV$ ...
2
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1answer
24 views

Absolutely continuous probability measures example

I was given the following definition: Then this example: It is said that $\mathbb P_1\ll\mathbb P_2$ , but I don't really see it.Please help.
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2answers
62 views

From distribution to Measure [duplicate]

I have been asked to create a new post with my question. So it is about starting from a distribution function and proving that we can always find a probability space. My attempt is this : So assume ...
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0answers
28 views

Relation between $L^1(T)$ and $L^1[0,1]$

I know the question may be too general, but I need to know if there is a way in which I could relate the spaces $L^1(T)$ (where $T=\{e^{2 \pi i x}: x \in [0,1]\}$ and we use the Lebesgue measure on ...
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0answers
29 views

From distribution function to probability measure

So assume we have a probability space $(\Omega, \mathcal{F}, P)$ and a random variable $X : \Omega \rightarrow \mathbb{R}^*$. We can derive from this a distribution $P_X$, and a distribution function ...
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1answer
32 views

Measurability of a Borel function

I need some help on the following proof. The claim is: Suppose $f:\mathbb{R}^k \to \mathbb{R}$ and $f \in B(\mathbb{R}^k)/B(\mathbb{R})$. i.e. Borel measurable. Let $X_1$,...,$X_k$ be random ...
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1answer
59 views

How do you show $\int \limits_{X \times Y} f(x,y)\, d\lambda < \infty$ if $\int \limits_{X} \int \limits_{Y} f(x,y) \,d\nu \,d\mu < \infty$?

Suppose $f: X\times Y \rightarrow [0,\infty]$ is a measurable function with respect to the product measure $\lambda$ ( $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$ are complete measure spaces). Suppose ...
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0answers
38 views

amazing boundedness problem from maximal function

Let $n\geq 2$. For any $M>1$, prove that there exists a constant $C_M>1$ such that for any ball $B$ in $\mathbb{R}^n$, if we denote $MB$ as the concentric ball of $B$ with $M$ times radius of ...
1
vote
1answer
34 views

Jordan Measure and Lebesgue Measure

The Jordan outer measure of $J^*(E)$ a set $E\subseteq \mathbb{R}$ is defined as infimim of $\sum_{i=1}^n (b_i-a_i)$ where $(a_i,b_i)$ are open intervals whose union contains $E$. The Jordan inner ...
1
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2answers
66 views

Measure of set of rational numbers

I find it difficult to understand why the 'size' of the set of rational numbers in an interval such as [0,1] is zero. I know that there are way more irrational numbers than rational numbers such that ...
0
votes
1answer
43 views

What is the Lebesgue measure of a following set?

This might be trivial for some of you. Let $E$ be a set defined by $ E= \{(x,y): a<\frac{x}{y}<b, c<\frac{y}{x}<d \}$. What is the Lebesgue measure of this set. Measure should be an ...
3
votes
2answers
54 views

Convergence in measure and $L_p$ implies product converges in $L_p$

This was given on an old comp as a true or false problem: If $1<p<\infty$, $|f_n|\leq 1$, $f_n\rightarrow f$ in measure, and $g_n\rightarrow g$ in $L_p$, then $f_ng_n\rightarrow fg$ in $L_p$. ...