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

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1answer
31 views

Does this sum of normally distributed random variables necessarily result in a continuous R.V?

Originally I had asked whether two continuous random variables can sum to a discrete random variable. More specifically, I am wonder whether, if we Let $X_n \sim \text{iid } N(0,\sigma_x^2)$ and $Y_n ...
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0answers
32 views

Real Analysis, Folland problem 3.2.14 The Lebesgue - Radon-Nikodym Theorem

Relevant background information: We say that two signed measures $\mu$ and $\nu$ on $(X,M)$ are mutually singular if there exists $E,F\in M$ such that $E\cap F = \emptyset$, $E\cup F = X$, $E$ is ...
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0answers
48 views

Understanding the concept of measurability of random variables

If a random variable $X$ is $\mathcal{F}_{t_0}$-measurable, where $\{ \mathcal{F} \} _{ t \geq 0}$ is an underlying filtration, does that mean that from the time $t_0$ onwards, the random variable $X$ ...
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0answers
36 views

Prove that this function is in $L^\infty$ with $\lVert g\rVert_\infty \le C$.

My professor used the following lemma in the proof that $L^1(X,\mu)^* = L^\infty(X,\mu)$ but left the proof as an exercise. Lemma. Assume that $(X,\mathcal A, \mu)$ is a measure space and $g \in ...
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1answer
43 views

Continuous Function on a Set With Content Zero

I am trying to prove a proposition about a continuous function over part of a compact set, and I have gotten stuck. The proof will be completed if I can verify the following: If $f$ is a continuously ...
7
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1answer
55 views

Never seen this notation before: $\int (y-f(x))^2 Pr(dx,dy) $

I have never seen an integral like this: $$\int (y-f(x))^2 Pr(dx,dy) $$ What is that? More precisely what is $Pr(dx,dy)$? And how is that integral defined? I found it in Elements of Statistical ...
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0answers
17 views

“Equidecomposable”: informal meaning

I am having trouble understanding the definition of the term "equidecomposable". Is it like two sets are split into many sets and then these many sets can be joined together to make either of the two ...
2
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2answers
41 views

Lebesgue Integral of a non negative piecewise function

Consider the function over [0,1] given by $f(x)= \begin{cases} 0 & x \in \mathbb{Q}\\ x & x \notin \mathbb{Q} \end{cases}$ In order to compute the Lebesgue integral of $f$ we need to find an ...
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0answers
59 views

How to understand $E(X\mid B)$ in the measure theory way

From undergraduate probability course, we learn $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ given $P(B)>0$. And we learn that if $(X,Y)$ has a joint density $f(x,y)$, we can calculate marginal density ...
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1answer
18 views

Real Analysis, Folland Problem 3.3.20 Complex Measures

Related definitions - A complex measure on a measurable space $(X,M)$ is a map $\nu: M\rightarrow\mathbb{C}$ such that i.) $\nu(\emptyset) = 0;$ ii.) if $\{E_j\}$ is a sequence of disjoint sets in ...
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1answer
42 views

Book Recommendation for Measure Theory in n-Space

What's a standard book on multidimensional measure theory? I'm aware of some books on functions of several variables, but they do not discuss measure theory or Lebesgue integration in space. Thanks. ...
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1answer
59 views

Integration over finite partition of integration domain

I think the title does not reflect my problem very well. Feel free to leave a comment with a more appropriate title. Let $f \in L^1([0,1])$. How do I prove there exists a partition of $[0,1]$ into ...
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2answers
35 views

Why a cover of a set exist?

In the definition of the measure, we have that $$m^*(E)=\inf\left\{\sum_{i=1}^\infty |Q_i|\mid E\subset\bigcup_{i=1}^\infty Q_i\right\}$$ where $Q_j$ are closed cube. My question is : Why for any $E$ ...
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0answers
29 views

Getting the independent variables from dependent variables. [duplicate]

This question is related to the solution in the answer here: Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent. Quick description of my problem: Let ...
2
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1answer
32 views

Prove that $\int g(x)dx=\int f(x)dx$.

Let $f:[0,b]\longrightarrow \mathbb R$ and $g:]0,b]\longrightarrow \mathbb R$ define as $$g(x)=\int_x^b\frac{f(t)}{t}dt.$$ Prove that $g$ is integrable and that $$\int g(x)dx=\int f(x)dx.$$ So ...
0
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1answer
22 views

Integration of a measurable function

Let $\phi(x)$ be a simple function. If $a_1, a_2, . . . . , a_n$ are the distinct values taken by $\phi$ and $A_i = [x : \phi(x) = a_i]$, then $\phi(x) =\sum_{i=1}^n a_i \Large {\chi}_{A_i}$ $(x)$ , ...
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0answers
37 views

Relation of absolute convergence to expected value.

Claim: If $X_n \overset{\text{a.s.}}{\longrightarrow} X$ then $\mathbf{E}[\lim_{n\to\infty}X_n] = \mathbf{E}[X]$. Question: Is this true? Below is a proof, but I'm worried that I made a mistake. ...
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0answers
29 views

Is the following modification of a martingale still martingale? [closed]

I have a following question. Let $Z$ be a Geometric Brownian motion, $\frac{dZ(t)}{Z(t)} = \omega dt + \sigma dW(t) $ For $\omega = -\frac{1}{2}\sigma^{2}$ one can proof that $Z$ is a martingale. ...
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0answers
47 views

Convolution of measures, why is the notation like this?

In both my book, and on Wikipedia they define convulution of two measures like this: $(\mu_1*\mu_2)(B)=\int_{\mathbb{R}^d}\mathcal{X}_B(x+y)d\mu_1(x)d\mu_2(y)$ It doesn't seem like a typo, but ...
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1answer
27 views

$\mu \ll m$ finite Borel implies $x \mapsto \mu(A + x)$ is continuous

Why is it true that if $\mu$ is a finite Borel measure on $\mathbf{R}$ which is absolutely continuous with respect to Lebesgue measure $m$, then $x \mapsto \mu(A + x)$ is continuous for any fixed ...
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0answers
35 views

Real Analysis, Folland problem 3.3.18 Complex measures

Related definitions - A complex measure on a measurable space $(X,M)$ is a map $\nu: M\rightarrow\mathbb{C}$ such that i.) $\nu(\emptyset) = 0;$ ii.) if $\{E_j\}$ is a sequence of disjoint sets in ...
1
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0answers
68 views

Why is linearity a requirement of a integral

I was reading Philip Protter's Stochastic Integration and Differential Equations textbook. He mentions that an operator, $I_X$, induced by $X$ should be linear to be called an integral. I have a ...
1
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1answer
35 views

Show that $l^p \subseteq l^q$ for $1 \leq p < q < \infty$

$$l^p = \{ (a_k)_{k \geq 1} : \sum \limits_{k=1}^{\infty} |a_k|^p < \infty \}$$ Since it is said $l^p \subseteq l^q$, I would have thought we have to show $$\sum \limits_{k=1}^{\infty} |a_k|^q ...
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0answers
20 views

Showing that $I(\xi_1,…,\xi_d)=0$

Let $\xi_1,...,\xi_d \in S^{d-1}$ and $Leb(B)=0$. We define $$I(\xi_1,...,\xi_d) = \int_0^\infty \cdots \int_0^\infty 1_B (r_1 \xi_1+\cdots+r_d \xi_d) \prod_{j=1}^d g(\xi_j,r_j) (r_j^2 \wedge 1) ...
3
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2answers
169 views

Applications of Dominated/Monotone convergence theorem

Consider a measure $\mu$ on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ where $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$. Consider the function $f: [0,\infty)\rightarrow ...
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2answers
30 views

Proving something is a norm

Let $a<b$ be real numbers and $X=C[a,b]$ be the space of continuous functions $f:[a,b] \rightarrow \mathbb R$ Prove that $||f||_1 = \sup _{t \in [a,b]} |f(t)|$ indeed defines a norm on $X$. The ...
3
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1answer
53 views

The partial derivative of a characteristic function (exercise).

Assume that you have a probability space $(\Omega, \mathcal{F},P)$ and a random varaible $X: (\Omega, \mathcal{F})\rightarrow(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$. Define the characteristic ...
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2answers
13 views

Regarding sigma fields and its subsets [closed]

There is a subset of sigma field $G_2$, say $G_1 \subset G_2$. $G_1$ is proven to be a sigma field. Does this necessarily imply that $G_1 = G_2$?
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1answer
19 views

Subsequence of $L^{2}(\Omega)$ - bounded sequence weakly * converging to a measure

I was reading a well available article in the internet: "THE COMPENSATED COMPACTNESS METHOD APPLIED TO SYSTEMS OF CONSERVATION LAWS by Tartar"; there at Page-266, it is written: "$L^{1}(\Omega)$ is ...
0
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1answer
34 views

Can one divide a set into subintervals [duplicate]

Sorry that the question title is unclear, I didn't know how to ask it. Take set $A \subseteq [0,1]$, measurable. Does there exist a sequence $x_1,x_2,\dots$ such that $\forall x_i$, \begin{align*} ...
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0answers
42 views

Integral Inequality (CDFs and PDFs)

Suppose I have a function $g \geq 0$ defined by $$g(x) = \int_{-\infty}^{x}f(t)\text{ d}t \geq 0\text{, }x \in \mathbb{R}\text{. }$$ I know for a fact that $g$ is continuous and nondecreasing. Is ...
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2answers
44 views

Is there a shorter proof for this variant of the Dominated Convergence Theorem?

I finally managed to proof this variant of the Dominated Convergence Theorem: Theorem (Variant of Dominated Convergence Theorem). Let $f, f_k: X \to \overline{\mathbb R}$ be $\mu$-measurable, $g, ...
3
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1answer
51 views

Structure of the $L_1$ space of measurable subsets of $[0,1]$

Let $\mathcal A$ be a Borel $\sigma$-algebra on $[0, 1]$, and let's introduce a metric on it by $$ d(A, B) = \lambda(A\mathbin\Delta B) \qquad \forall A,B\in \mathcal A $$ where $\lambda$ is the ...
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0answers
17 views

Invariant $\sigma-$ field of double infinite stationary process

We know any stationary process ${(X_n)}_{n \in \mathbb{Z}}$ can be represented as $X_n(\omega)=X(\phi^n\omega)$. where we take the shift $\phi$ on the canonical space of the process and $X$ maps a ...
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0answers
41 views

If $X$ does not have a density, what is $\int x \ d X(P)$?

Let $X$ have distribution function $$F(x) = 1_{(-\infty,0)} e^x + 1_{[0,\infty)} (1 - e^{-x}/3).$$ My questions are: If a general distribution function is not a nice continuous function, does X ...
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2answers
53 views

Let $\mu_n$ be a sequence of finite measures on space $(X,M)$ and $\forall E \in M, \lim_{n \to \infty }\mu_n(E)=\mu(E)< \infty $..

Let $\mu_n$ be a sequence of finite measures on space $(X,M),M-\text{ sigma algebra on X}$ and $\forall E \in M, \lim_{n \to \infty }\mu_n(E)=\mu(E)< \infty $ and let $f$ be a bounded function. ...
1
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0answers
22 views

Density of a measure with respect to another measure

Consider $\mathbb{P}, \mu$ measures on the measurable space $(\Omega, \mathcal{F})$. Suppose $\mathbb{P}$ has density $p$ with respect to $\mu$. Let $A \in \mathcal{F}$. Statement: ...
2
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1answer
20 views

$f$ real-valued function that dies of in infinity but $f^p$ not integrable for any $p$.

Is there a positive continuous function on $\mathbb R$ such that $f(x) \to 0$ as $x \to \pm \infty$ but $f^p$ not integrable for any $p>0$?
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2answers
35 views

Does the function $f(x)=\frac{1}{\sqrt x}$ belong to $L^p( \mathbb N , P(\mathbb N), \mu),p=1,2,\infty?$

Does the function $f(x)=\frac{1}{\sqrt x}$ belong to $L^p( \mathbb N , P(\mathbb N), \mu),p=1,2,\infty?$ $\mathbb N$- set of natural numbers, $P(\mathbb N)$- the partitive set of natural numbers. I ...
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0answers
39 views

Measurability of the set where sample path is continuous

Let $(\Omega,\mathscr F, \mathbb P)$ be a probability space and let $(X_t)_{t>0}$ be a collection of random variables such that $X_t:\Omega\to\mathbb R$ is $\mathscr F$-to-Borel measurable. Fix ...
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0answers
18 views

Almost everywhere convergence plus convergence of integrals imply converence in L^1 [duplicate]

Consider non-negative measurable functions $f, f_n$ on a measure space $(X, \mathcal A, \mu)$. How does one show that $f_n \to f$ almost everywhere and $\int f_n d\mu \to \int f d \mu$ imply $f_n \to ...
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1answer
28 views

Questions on symmetric difference of events

From a comment on my math overflow question: No, $P(A\bigtriangleup B)=0$ means $A$ and $B$ are essentially the same except in situations that almost surely do not happen. $P(A)=P(B)$ says much ...
2
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1answer
53 views

Given that $f_n \to f$ in $L^1(\Omega)$, $\mu(\Omega )=1$ and $ \|f_n\|_2^2 \leq M$, show $ \|f\|_2^2 \leq M$.

Given that $$\int_{\Omega} |f_n -f | \, d \mu \to 0,$$ $\mu(\Omega )=1$ and $ \|f_n\|_{L^2}^2 \leq M$, show $ \|f\|_{L^2}^2 \leq M$. Attempt: Note first that $f_n \in L^1(\Omega)$ since $$\|f_n ...
4
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1answer
44 views

$\int_{\Omega} |f_n-f||f_n| \, d \mu \to 0$ if $f_n \in L^1(\Omega)$, $f_n \to f$.

Suppose $f_n \to f$ in $L^1(\Omega)$ where $\mu(\Omega)=1$. Suppose $$\int_{\Omega} |f_n| \, d\mu \leq M$$ for all $n$. Is there a way to show that the integral $$\int_{\Omega} |f_n-f||f_n| \, d ...
1
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1answer
28 views

Let $(X, M, \mu )$ be a space with measure. $f:X \to \mathbb R \text{ and } f\in L^1(X).$ ..

Let $(X, M, \mu )$ be a space with measure. $f:X \to \mathbb R \text{ and } f\in L^1(X).$ Prove that for all $\epsilon > 0$ that there exists $\delta > 0$ such that for $E \in M$, $\mu(E)< ...
2
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3answers
48 views

Continuous function $f$ on $\mathbb R $ such that $f \notin L^1 (\mathbb R)$ but $f \in L^1([a,b]), a< b $

Give an example of a continuous function $f$ on $\mathbb R $ such that $f \notin L^1 (\mathbb R)$ but $f \in L^1([a,b]), a< b $ If $f \in L^1([a,b]), a< b$ that would mean that ...
0
votes
0answers
19 views

If $E \subset [0,1]$ satisfies, for any $I \subset [0,1]$, $m(E \cap I) \geq \frac{1}{2}m(I)$, then $m(E)=1$

If $E \subset [0,1]$ satisfies, for any $I \subset [0,1]$, $m(E \cap I) \geq \frac{1}{2}m(I)$, then $m(E)=1.$ I'm aware this post exists elsewhere, say, here but what I don't understand is why we ...
0
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1answer
38 views

Proving that $f(x)=\frac{1}{x^2 \ln x} $ is Lebesgue measurable on $(2, + \infty)$

I have that a set $E$ is Lebesgue measurable if the outer measure: $$\mu^*(E)=\inf_{I_1,...,I_n} \mu (I), E \subseteq I_1 \cup I_2 ,...\cup I_n , I_i-\text{intervals}$$ satisfy the three properties ...
4
votes
1answer
33 views

Proving weak convergence of random probability measures

I don't understand the following as I read along a proof in a paper: We denote by $\mathcal{P}({M})$ the space of probability measures on a metric space $M$, equipped with the weak topology. ...
1
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0answers
32 views

Symmetric difference and approximation of measure [duplicate]

Let $\scr{A}$ be an algebra of subsets. Let $(\Omega, \sigma(\mathscr{A}), P)$ be a probability space. Then for each $B \in \sigma(\scr{A})$ and $\epsilon > 0$, there exists $A \in \scr{A}$ such ...