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

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Is an infinite product probability space compatible with an almost surely statement?

My question pertains to the following Theorem which can be found here. Theorem (Existence of product measures): Let $A$ be an arbitrary set and for each $\alpha \in A$ let ...
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34 views

If $X_n$ are independent random variables, then does $\sum_n \mathbb{E}(X_n)=\mathbb{E}(\sum_n X_n)$?

If $X_n$ are independent random variables, then does $\sum_n \mathbb{E}(X_n)=\mathbb{E}(\sum_n X_n)$? This is not a homework problem but rather a question I had. If it is not true, what are the ...
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1answer
28 views

$\int f(x) dm = \int f(x + a) dm$

$f: \mathbb{R} \rightarrow [0,\infty]$ is Lebesgue-measurable and $m$ is the Lebesgue measure. Show that for $a\in \mathbb{R}$ $$ \int f(x)dm = \int f(x+a) dm.$$ I know Lebesgue measure is ...
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1answer
11 views

Support, measure, function

I have a question about a support. Let $m$ be Lebesgue measure on $\mathbb{R}^{d}$ and $f$ be a continuous function on $\mathbb{R}^{d}$. We define $\mu(A):=\int_{A}|f|dm,\,A{\rm \,: ...
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1answer
31 views

Analytic skills in applied math

I am an unexperienced undergraduate student just took few basic math classes. And I have found analysis classes really interesting, like basic analysis, measure theory and functional analysis, and ...
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11 views

Total Variation of a Signed Measure

In Royden's Real Analysis the total variation $|\mu|$ of a signed measure $\mu$ is defined to be $$|\mu|(E) := \sup\sum_{k=1}^n |\mu(E_k)|,$$ where the supremum is taken over all finite disjoint ...
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1answer
9 views

A property of conditional expectation

Given a probability space $(X , \mathcal{M} , m)$ and $\mathcal{A}$ is a $\sigma$ sub algebra of $\mathcal{M}$. Let $\mathbb{E}$ be the condition expectation given $\mathcal{A}$. Given $f$ is an ...
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12 views

Conditional expectation, conditioning on a random variable [duplicate]

I am asked to show that if $X,Y$ are integrable, and $E[X| Y]=Y$ and $E[Y|X]=X$ almost surely, then $X=Y$ almost surely. Moreover, is the first equality sufficient for $X=Y$ almost surely? My attempt ...
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1answer
27 views

borel measurable and measurable

Let $(\Omega,A,\mu)$ be a measure space and let f:$(\Omega,A)$$\to$$(\mathbb{R},\mathbf{B})$ be a nonnegative measurable function and define define the graph ...
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14 views

Monotone measureable function that is bounded on domain [on hold]

Given $( \mathbb X , \mathscr A, u)$, Given a measurable, monotone function $f$ defined on a bounded set, Show that this function is $u$-integrable.
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Measure-preserving maps on probability spaces

A professor posed me a problem a few days ago, and I have not been able to find an answer to it. Let $(X_1,\Sigma_1,\mu_1)$ and $(X_2,\Sigma_2,\mu_2)$ be probability spaces. Suppose the following ...
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1answer
19 views

Relationship between $m(E\backslash F) < \epsilon$ and $m(E)-m(F) < \epsilon$

This may be a fairly easy question but I just want to make sure there are no strange counterexamples. Suppose $E$ and $F$ are measurable sets such that $F \subset E$. Now suppose $m(E \setminus F) ...
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0answers
18 views

Kolmogorov zero-one law in continuous time?

Let $(X_t : t \geq 0)$ be a stochastic process. Is it necessarily the case that $$P (\limsup_{t \geq 0} X_t \leq a) \in \{ 0,1\}$$ as it is in discrete time? If some conditions are needed on the ...
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0answers
23 views

Tonelli-Fubini Theorem for two copies of $\mathbb{N}$ with counting measure

I just learned the Tonelli-Fubini Theorem and I was wondering what does it say for two copies of $\mathbb{N}$ when considering the counting measure. And what is the difference when we consider one ...
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1answer
41 views

Show that the measure is zero

I am asked to show that the $2-$dimensional Lebesgue measure of the graph of a continuous real function is zero. Could you give me some hints how I could show it??
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11 views

Functional representation of adapted jointly measurable stochastic processes

Let $X_t : \Omega \to E, \ t \geq 0$ be continuous-time stochastic process with (Polish) state space $E$ and canonical filtration $\mathcal{F}_t := \sigma(X_u \ | \ u \leq t)$. Let $Y_t : \Omega \to ...
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7 views

Continuity of signed measure

I want to prove that, given a signed measure $\nu$ on $(X,M)$, continuity from above and from below continues to hold. What I'm uncertain about is any difference in the proof for a signed measure ...
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1answer
23 views

On convergence a.e and convergence measure

I have a question. First, I know that convergence in measure of a sequence of functions $f_n$ is different than convergence a.e., wich means there are sequences that converge in measure but not a.e. ...
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1answer
16 views

Sigma-algebra generated by a sigma-algebra

I know intuitively that $$\sigma(\sigma(\theta))=\sigma(\theta),$$ where $\theta$ is the class of all open sets in $\mathbb{R}^\mathrm{k}$. But why? How can I prove it? Also, is ...
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18 views

Polynomial density in $L^p (\mathbb{R},\mu)$

I wanna check a necessary and sufficient condition for a Radon measure witch have the moments of all orders, to say that polynomials are dense in $L^p (\mathbb{R},\mu)$. Or just a paper or an article ...
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1answer
54 views

$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? [duplicate]

$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? My feeling is that this is not necessarily true. But cannot come up with an example. Can someone provide ...
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26 views

How to show function is measurable if and only if each component is

Let $(X,\mathcal{M})$ be a measurable space and $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ be a measurable space of Borel sets on the real line. Let $f_i:X\rightarrow \mathbb{R}$ be given for ...
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1answer
14 views

Lebesgue measurable functions

Let $f(x)=x^{-1/2}$ if $0<x<1$ and $f(x)=0$ otherwise. Is f a Lebesgue measurable function from $\mathbb{R} \rightarrow [0,\infty]$? If it is a Lebesgue measurable function how can I show it?
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1answer
16 views

Different ternary representations

I just picked up the subject of ternary expansions (actually I'm trying to gain an understanding of the cantor set for measure theory) so my knowledge is still extremely weak but I just a quick ...
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19 views

On Compact and Measurable Sets with Positive Length

Greetings fellow Mathematics enthusiasts! The following two-part problem is giving me trouble, and I was hoping someone could help me solve it. It is coming from Terrence Tao's Introduction to Measure ...
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7 views

Showing that a precursor to the packing measure is not a measure

I am trying to prove the highlighted sentence. What countable dense sets should I consider? and how am I trying to prove this is not a measure?
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0answers
43 views

Complete additivity of set functions

I need help with the following two statements of measure theory, for which I am learning for at the moment: For any non-empty set $T \subseteq [0, \infty]$ of non-negative numbers (allowing ...
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0answers
21 views

Borel measure and Riesz measure

Let $\mu$ be a Borel measure on $\mathbb{R}^n$ s.t. $\mu(K)< \infty$ for all compact $K$. Show that $\mu$ is the restriction of some Riesz measure on $\mathcal{B}$. I try to prove it using the ...
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0answers
31 views

Characterization of Lebesgue measure based on translation invariant

I am trying to solve a problem about characterization of Lebesgue measure. Let $(\mathbb{R}^n, \mathcal{B}, \mu)$ be a Borel measure space whose measure $u$ is translation invariant and exist a set ...
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3answers
77 views

Prove that $\|x+y\|_{\infty} \leq \|x\|_{\infty} + \|y\|_{\infty}$.

Suppose $\left(X, \Sigma, \mu \right)$ is a measure space and $x,y \colon X \longrightarrow \mathbb{R}$ are random variables. We define $$\|x\|_{\infty} := \inf_{A \subseteq \Sigma, \mu(A)=0} ...
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1answer
12 views

a bounded function is converge in measure, then its limit is also converge

If a series function, ${f_n} \rightarrow f $ in measure $\mu$, and $|f_n| \leq M$, how to show that $|f| \leq M$? My instructor gave a hint as follows, but I do not believe the first inequality ...
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36 views

Proving that any measure is the sum of a semi-finite measure and a measure which takes either 0 or infinity.

I need help proving that for a measure space (X, A, u) that u can be written as the sum of a semi-finite measure and a measure that takes on the values 0 and infinity. The second measure, u_i I ...
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1answer
43 views

$\int_X f(x)\,d\mu\,\,$ exists iff $\,\,\int_X \lvert \,f(x)\rvert\,d\mu\,\,$ does

I know that, for a domain of finite measure $X$, provided that $f$ is measurable, each of the Lebesgue integrals$$\int_X f(x)d\mu\quad\text{ and }\quad\int_X |f(x)|d\mu$$exists if and only if the ...
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1answer
15 views

Measure of sum of sets of “Cauchy” sequence bounded?

Let $\{A_n\}_n$ be a sequence of sets of a $\delta$-ring $\mathfrak{M}$ of measurable sets with finite Lebesgue measure. Let us suppose that $$\forall\varepsilon>0\quad\exists ...
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1answer
30 views

Covering Class and Describing Outer MEasure for General Measures

I am uncertain if my description is correct, but I describe the measure in a piecewise type fashion. In general, $\mu_{\lambda}^*(A) = \infty$, if $A = X$ or $A$ uncountable. $\mu_{\lambda}^*(A) = ...
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1answer
40 views

Prove that $\mathcal B(\mathbb R)\times \mathcal B(\mathbb R)\subseteq \mathcal B (\mathbb R^2)$

I need to prove that $$\mathcal A(\mathcal B(\mathbb R)\times \mathcal B(\mathbb R))= \mathcal B (\mathbb R^2)$$ Where $\mathcal B$ is the generated Borel algebra and $\mathcal A$ is the generated ...
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0answers
64 views

measure theory exercise (verification)

Hi I found the following exercise in the Dudley's book and I'd like to see if my answer is correct; the last part is what I'm not entirely sure, since I'm not completely familiar with this kind of ...
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1answer
27 views

absolute continuity - Dirac measure with respect to gaussian measure [duplicate]

Let $a \in \mathbb{R}$ and Dirac measure $\delta_a (A) = 0$ if $a \notin A$ and $\delta_a(A) = 1$ if $a \in A$, and let $\mu_1$ be the one-dimensional gaussian measure. Let $\mu$ and $\nu$ be two ...
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1answer
37 views

Name for LDC: Lebesgue?

Is there also a name associated to the Lebesgue dominated convergence theorem like Beppo-Levi or Fatou? Would Lebesgue be reasonable? Who did originally prove it?
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1answer
20 views

Approximation in $L^2(\Omega)$

I want to prove that if $f_n\to f$ in $L^2(R)$ then $f_n(X)\to f(X)$ in $L^2(\Omega)$ for each random variable X. I think of using the dominated convergence theorem, having the puntual convergence, ...
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39 views

integral with respect to Dirac measure

Let $\delta_a(A)$ be the Dirac measure, that is $\delta_a(A) = 0$ if $a \notin A$ and $\delta_a(A) = 1$ if $a \in A$ and $\phi : \mathbb{R} \rightarrow \mathbb{R}$ a bounded Borel function. What does ...
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1answer
38 views

What does it mean m(dx), where m is Lebesgue measure?

Let a $\in \mathbb{R}$, $\phi_n : \mathbb{R} \rightarrow \mathbb{R}_+$, $\phi_n (x) : = \frac{n}{\sqrt{2 \pi}} e^{\frac{-n^2 x^2}{2}}$, $n \geq 1$ and let $\mu_n (d x) : = \phi_n (x - a) \lambda (d ...
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38 views

Measurability and a integral

I need to calculate $\lim_{n\rightarrow\infty}\int^{\infty}_{0}\frac{cos(\frac{x}{n})}{2^x}d\lambda(x)$ and show that the integral makes sense for every $n$. My approach so far: Let ...
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1answer
31 views

Approximation theorem from measure theory

Let $a$ be an algebra, $\mu_0$ a pre-measure on it, and $\mu$ be a measure on the generated $\sigma$-algebra. Let $E \in \sigma (a) $, such that $\mu (E) <\infty $. Show that $\forall ...
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2answers
72 views

What is the motivation to build measure theory?

I started reading about measure theory on wikipedia, and downloaded some PDFs, but they all start defining things that I can understand, but can't imagine the motivation to define these things. ...
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1answer
24 views

Definition of a $\sigma$ - finite set

I know the definition of a $\sigma$-finite measure. But I found a problem in which it asks to show a particular set is $\sigma$ finite? But what is a $\sigma$ finite set? This is the problem I found. ...
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1answer
51 views

If $X_n \rightarrow X$ almost surely then $f(X_n) \rightarrow f(X)$ almost surely

Proof: If f is continuous and $X_n \rightarrow X$ almost surely, then $f(X_n) \rightarrow f(X)$ almost surely. Thats the only information I have. Does this only hold if the measure on the target ...
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1answer
18 views

$L^p$ spaces and converging sequence in this space

I have a question about $L^p$ space which I kan not solve it could u plz help me: let $(\Omega,A, \mu)$ be a measure space and let $1<p<\infty$.let $f_n$:$\Omega$ $\to$$\mathbb{C}$ be a ...
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1answer
35 views

Lebesgue integral and anti-derivative

For which Lebesgue measures the Lebesgue integral of a differentiable function over a Euclidean space or an orientable manifold coincides with its anti-derivative? For example, can we find the class ...
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2answers
38 views

Why is the outer measure of the set of irrational numbers in the interval [0,1] equal to 1?

Just learned Lebesgue outer measure from Royden's Real Analysis. Let me give my proof. First, let $A$ be the set of irrational numbers in [0,1]. So $A\subset [0,1]\Rightarrow m^*(A)\le ...