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

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Understanding conditional independence of two random variables given a third one

I am reading the Wikipedia article on conditional independence. There seems to be Two definitions for conditional independence of Two random variables $X$ and $Y$ given another one $Z$: Two random ...
5
votes
1answer
486 views

Measure Theory and Integrals of Characteristic Functions

Given two sets of finite measure in $\mathbb{R}$ say, $E$ and $F$, and their characteristic functions $\chi_E$ and $\chi_F$, can somebody show that $\chi_E\ast\chi_F(x)$ (the convolution) is a ...
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vote
1answer
255 views

measurability of limit superior for functions

Suppose $(X,\mathcal {M})$ is a measurable space. For function $f(r,x): \mathbb R\times X\to\mathbb R$, suppose each r-section of it is $\mathcal M$-measurable. For constant $R\in\mathbb R$, do we ...
5
votes
1answer
118 views

Existence of a measure that preserves a given mapping

Let $(X, d)$ be a compact metric space and let $T:X \to X$ be a continuous mapping. Now, does there exist a probability measure $\nu$ such that $T_* \nu = \nu$ (the first thing is the image measure)? ...
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2answers
181 views

measure theory question

hello im having trouble showing the following: let $u$ be a positive measure. if $\int_E f\, du= \int_E g\, du$ for all measurable $E$ then $f=g$ a.e. i was trying to argue by contradiction: if ...
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1answer
275 views

Russell's paradox and the foundations of measure theory

Measure theory was established on naive set theory(Not totally sure). But after Russell discovered the paradox named by him, set theory was reconstructed in the sense of axiomatization. My question ...
8
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3answers
373 views

Is $SO_2$ an amenable group?

In S. Wagon's "The Banach-Tarski Paradox," amenable groups are defined on p. 12 as follows: [amenable] groups bear a left-invariant, finitely additive measure of total measure one that is defined ...
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1answer
418 views

question about total variation (measure theory)

suppose $v$ is a signed measure on $(X,M)$ and $E\in M$ how do i go about showing that $|v|(E)=sup\{\sum_1^n |v(E_j)|: E_j\cap E_i=0 \forall i\neq j, \cup_1^n E_j=E\}$ sorry it took me awhile to fix ...
4
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1answer
413 views

A measurable function on an atom is almost everywhere constant

Let $f \in m(\Omega,\mathcal{F})$, i.e. $f \mapsto [-\infty,\infty]$ and let $A \in \mathcal{F}$ be an atom. Prove that $f$ is almost everywhere constant on A: there exists $k \in [-\infty,\infty]$ ...
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2answers
1k views

Is there a function with infinite integral on every interval?

Could give some examples of nonnegative measurable function $f:\mathbb{R}\to[0,\infty)$, such that its integral over any bounded interval is infinite?
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1answer
249 views

Absolute continuity of measures

Suppose $u$ and $v$ are measures on a measurable space $E$. Further suppose $u$ is finite and absolutely continuous with respect to v ($v(S)=0 \implies u(S)=0$). The problem is: Show that $\forall ...
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1answer
202 views

Measurability of a function

I am having a hard time understanding a condition in building a measurable function. This is an example of a measurable function from this book (p. 55). Quoting verbatim: Consider $R_n(\omega)$, ...
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votes
2answers
625 views

Why events in probability are closed under countable union and complement?

In probability, events are considered to be closed under countable union and complement, so mathematically they are modeled by $\sigma$-algebra. I was wondering why events are considered to be ...
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3answers
830 views

Question on the notion of a $\sigma$-algebra generated by a function

I've started learning about measure theory and I'm trying to get some intuitive grasp of the basic concepts. This is only succeeding partially so far. There is an exercise which I don't quite ...
7
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1answer
3k views

Interpretation of sigma algebra

My question is how to interpret sigma algebra, especially in the context of probability theory (stochastic processes included). I would like to know if there is some clear and general way to interpret ...
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2answers
1k views

Limit a.e. of a sequence measurable functions is measurable

I'm having trouble showing the following: If $f_n$ is a sequence of measurable functions such that $f_n$ converges to $f$ almost everywhere, then $f$ is measurable. I was thinking of using ...
7
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1answer
856 views

Relation between Borel–Cantelli lemmas and Kolmogorov's zero-one law

I was wondering what is the relation between the first and second Borel–Cantelli lemmas and Kolmogorov's zero-one law? The former is about limsup of a sequence of events, while the latter is about ...
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1answer
276 views

Building a Bernoulli sequence where finite patterns repeat infinitely often

As a step in proving that the union of intervals $B_E \subset [0,1]$ (where $E$ is the set of infinite Bernoulli sequences, e.g. 0.0110...., such that some finite pattern repeats infinitely often) is ...
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2answers
496 views

Measure from on product $\sigma$-algebra to on component $\sigma$-algebras

This is inspired by Carl Offner's reply to one of my previous questions and my previous question about marginal and joint measures. Given a measure $\mu$ on product $\sigma$-algebra $\prod_{i \in I} ...
3
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1answer
196 views

Further questions about from product $\sigma$-algebra to component ones

This is a continuation of my previous question and inspired by Arturo Margidin's reply. Suppose there are a collection of measurable spaces $(X_i, \mathbb{S}_i), i \in I$. Let $\mathbf{X}=\prod_{i\in ...
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1answer
779 views

How to understand marginal distribution

Given a random vector $X: (\Omega, \mathbb{F}, P) \rightarrow (\prod_{i \in I} S_i, \prod_{i \in I} \mathbb{S}_i)$, is each component variable $X_i, \forall i \in I$ of the random ...
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3answers
4k views

Under what condition we can interchange order of a limit and a summation?

Suppose f(m,n) is a double sequence in $\mathbb R$. Under what condition do we have $\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$? ...
3
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1answer
670 views

Is product of measurable subsets measurable wrt infinite product $\sigma$-algebra?

This is a continuation of my previous question. I admit I still don't fully understand the concept of infinite product of $\sigma$-algebras. Let $(E_i, \mathbb{B}_i)$ be measurable spaces, where $i ...
3
votes
2answers
629 views

From $\sigma$-algebra on product space to $\sigma$-algebra on component space

Given a $\sigma$-algebra on a Cartesian product of a collection of sets, do there always exist a $\sigma$-algebra on each set, so that their product $\sigma$-algebra on their Cartesian product will be ...
2
votes
1answer
366 views

Questions about measurable mapping and product sigma algebra

Suppose there are three measurable spaces $(\Omega, \mathbb{F})$, $(S_i, \mathbb{S}_i), i=1,2$, and two measurable mappings $f_i: \Omega \rightarrow S_i, i=1,2$. Is the mapping $f$ defined ...
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2answers
495 views

Pull the teeth out of Lebesgue integration

In Lebesgue integration we usually approximate the function we want to integrate with step-functions on measurable sets. How much "power" do we take away if we require that the step functions are on ...
3
votes
2answers
598 views

What does it mean to say a function is differentiable with respect to lebesgue measure?

What does it mean to say a function is differentiable with respect to the lebesgue measure almost everywhere. A definition would be helpfull. Do I need to learn about the Radon Nikodym derivative to ...
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3answers
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Is there such an inequality between product and integral for functions

Given a measure space $(\Omega, \mathbb{F},\mu)$ and any two measurable real-valued functions $f,g$ defined on $\Omega$, I was wondering if there is an inequality like $$ \int_\Omega |f*g| d\mu \leq ...
5
votes
1answer
896 views

G-delta set with the same Lebesgue outer measure

Statement: If E is a bounded set of real numbers, there exists a G-delta set G such that G contains E and has the same Lebesgue outer measure with E. I completed the proof for the case that E is ...
4
votes
1answer
566 views

How to understand joint distribution of a random vector?

Given a random vector, what are the domain, range and sigma algebras on them for each of its components to be a random variable i.e. measurable mapping? Specifically: is the domain of each ...
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votes
2answers
373 views

Questions about product measure

On Planetmath, product measure is roughly defined as follows: Let $(E_i, \mathbb{B}_i, u_i)$ be measure spaces, where $i\in I$ an index set, possibly infinite. When each $u_i$ is totally finite, ...
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1answer
536 views

Integral of measurable spaces

If for each $t\in I=[0,1]$ I have a measurable space $(X_t,\Sigma_t)$, is there a standard notion which will give a measurable space deserving to be called the integral $\int_I X_t\,\mathrm d t$? ...
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2answers
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Infinite product of measurable spaces

Suppose there is a family (can be infinite) of measurable spaces. What are the usual ways to define a sigma algebra on their Cartesian product? There is one way in the context of defining product ...
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1answer
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Nonmeasurable set with positive outer measure

It is well-known that any set $E \subseteq \mathbb{R}$ with positive outer measure contains a nonmeasurable subset $V$. I know that $0 < m^*(V) \le m^*(E)$. Nevertheless, my question is the ...
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1answer
291 views

Uniform continuity question

Let $f$ be a measurable function on $\mathbb{R}$ satisfying $\displaystyle \int\mid f(x)\mid dm(x)< \infty$, where $m$ is the Lebesgue measure on the Borel subsets of $\mathbb{R}$. Show that the ...
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1answer
2k views

How to show that $L^p$ spaces are nested?

Suppose $1<p_1<p_2<\infty$, then show $L^{p_1}[a,b] \supset L^{p_2}[a,b]$. I was able to show $||f||_{p_1} \le ||f||_{p_2} (b-a)^{1/p_1 - 1/p_2}$ but I'm not sure how to proceed from here.
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1answer
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Does this induce a proper probability distribution on the space of covariance matrices?

Say I define a probability distribution on $P$ dimensional symmetric matrices such that the diagonals are strictly positive but the off diagonals are unrestricted (except for the symmetry constraint). ...
8
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1answer
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integral of the cantor function

consider the ternary cantor set C, and the asscoiated cantor function f, and the associated Lebesgue-Stieltjes measure u. what is the integral of f over all of R with respect to u? my attempt: i ...
5
votes
1answer
655 views

Is there a compact subset of the irrationals with positive Lebesgue measure?

Does there exist $K \subseteq \mathbb{R} \backslash \mathbb{Q}$ such that $K$ is compact, and has Lebesgue measure greater than $0$? As I have been trying to think of examples, I suspect that any ...
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2answers
943 views

Confused about Cantor function and measure of Cantor set

OK, so I know that the Lebesgue measure of the ternary Cantor set is $0$. However, in class the prof briefly mentioned that if we build a Lebesgue-Stieltjes measure $\mu$ out of the Cantor function, ...
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1answer
230 views

Show that this function is not continuous except on a set of measure zero

Let $\{r_n\}_{n\in\mathbb{N}}$ be a enumeration over the rationals Let $$g(x)=\sum_1^\infty \frac{1}{2^n} \frac{1}{\sqrt{x-r_n}} \chi_{(0,1]}$$ where $$\chi_{(0,1]} = \left\{\begin{array}{ll} ...
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3answers
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The sum of an uncountable number of positive numbers

Claim:If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$ such that $\sum_{\alpha\in A}x_\alpha<\infty$, then $x_\alpha=0$ for all but at most countably many ...
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4answers
298 views

Limits and Measure Theory

How would I evaluate these limits? $$ \lim_{n \to \infty} \int_0^\infty \frac{n}{1+(nx)^2} \ dx$$ and $$ \lim_{n \to \infty} \int_0^\infty \frac{(1+(nx)^2)}{(1+nx^2)^n} \ dx$$
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2answers
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Lebesgue measurable but not Borel measurable

I'm trying to find a set which is Lebesgue measurable but not Borel measurable. So I was thinking of taking a Lebesgue set of measure zero and intersecting it with something so that the result is not ...
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1answer
455 views

Is there a measurable set $A$ such that $m(A \cap B) = \frac12 m(B)$ for every open set $B$?

Is there a measurable set $A$ such that $m(A \cap B)= \frac12 m(B)$ for every open set $B$? Edit: (t.b.) See also A Lebesgue measure question for further answers.
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2answers
218 views

How do different notions of “distribution” relate to one another?

In reading "Real Analysis: Modern Techniques and Their Applications" (Folland), I've come across a few different notions of "distribution" or "distribution functions." The distribution function of a ...
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1answer
248 views

Visualization of 2-dimensional function spaces

As a follow-up question to what is the norm measuring in function spaces I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal ...
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3answers
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What is the norm measuring in function spaces

In spatial euclidean vector spaces norm is an intuitive concept: It measures the distance from the null vector and from other vectors. The generalization to function spaces is quite a mental leap (at ...
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2answers
121 views

Why are signed and complex measures typically not allowed to assume infinite values?

In a number of real analysis texts (I am thinking of Folland in particular), three different kinds of measures are defined. Positive measures: Take values in $[0, +\infty]$ Signed measures: Take ...
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Does $\int_{\mathbb R} f(x)x^n dx = 0$ for $n=0,1,2,\ldots$ imply $f=0$ a.e.?

Let $f(x)$ be a real-valued function on $\mathbb{R}$ such that $x^nf(x), n=0,1,2,\ldots$ are Lebesgue integrable. Suppose $$\int_{-\infty}^\infty x^n f(x) dx=0$$ for all $n=0,1,2,\ldots. $ Does it ...