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

learn more… | top users | synonyms (1)

0
votes
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
votes
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
votes
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 $\phi(x) =\sum_{i=1}^n a_i \Large {\chi}_{A_i}$ $(x)$ , ...
0
votes
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. ...
1
vote
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. ...
1
vote
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 ...
1
vote
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 ...
0
votes
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
vote
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
vote
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 ...
1
vote
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
votes
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 ...
1
vote
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
votes
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 ...
0
votes
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 $G_1 = G_2$?
0
votes
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
votes
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*} ...
1
vote
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 ...
0
votes
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
votes
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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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
vote
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
votes
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 $p>0$?
1
vote
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 ...
3
votes
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 ...
0
votes
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 ...
-1
votes
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
votes
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
votes
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
vote
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
votes
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
votes
1answer
37 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
vote
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 ...
0
votes
2answers
46 views

Solving these types of integrals, using Monotone convergence theorem and Dominated convergence theorem.

I'm allowed to use these two theories and obviously the standard techniques when solving integrals. $$\lim_{n\to \infty } \int_{0}^{1}\frac{n^{\frac{1}{2}} x \ln x}{1+n^2x^2}dx$$ I did a similar ...
0
votes
0answers
23 views

Compactness of the outer measure

I have maybe a naive question about compactness of outer measure (or completion). Let $(E,\mathcal B(E))$ be a Polish space, and $\mathcal M_b(E)$ the bounded Radon measure on $E$. Assume that a ...
0
votes
1answer
22 views

Show: $\int_{\Omega} f d\mu =\int_{]0,\infty[} \mu(E_t) d\lambda_1(t)$

I have troubles understanding one step in the solution for this task: Let $(\Omega,\mathcal{A},\mu)$ be a $\sigma$-finite measure and $f:\Omega \rightarrow [0,\infty]$ be a measurable function. Let ...
5
votes
1answer
51 views

uniform boundedness principle for $L^{1}$

i read this theorem from V.I.Bogachev vol 1 Measure Theory. A family $\mathcal{F}\subset L_{1}(\mu)$,where the measure $\mu$ takes values in $[0,+\infty]$, is norm bounded in $L_{1}(\mu)$ precisely ...
0
votes
0answers
30 views

$\mathbb{R}=\cup_{i=1}^{\infty}A_{i}$, all $A_{i}$ Borel subsets of $\mathbb{R}$ and only contain a finite number of rational numbers?

Can $\mathbb{R}$ be written as a countable union of set $A_{i}$ such that all $A_{i}$ are Borel subsets of $\mathbb{R}$ and only contain a finite number of rational numbers? Moreover, are the ...
0
votes
1answer
20 views

Integrate step function and characteristic function

Let be $f_{k} = \frac{1}{k} \mathbb{1}_{[-k,k]}$ and $f_k: \mathbb{R} \rightarrow \mathbb{R}$. How i can show that $f_{k}$ is integrable $\forall k\in \mathbb{N}$? and HOW to compute this? $f_{k}$ ...
2
votes
1answer
45 views

$\sigma$-algebra generated by random variable : Show that if $\sigma(X)=\sigma(Y)$ then $\sigma(X+Y)\subseteq \sigma(X)$

Let $(\Omega,\mathcal{F},P)$ be a probability space and $X$ be a random variable. The $\sigma$-algebra generated by $X$ is defined as $$\sigma(X):=\{X^{-1}(B)\; | \; B\in B_{\mathbb{R}}\}$$ where ...
2
votes
0answers
64 views

Simple exercise measure theory/$\sigma$-algebras

Is this right? Q: Find an infinite collection of subsets of $\mathbb{R}$ that contains $\mathbb{R}$, is closed under the formation of countable unions, and is closed under the formation of countable ...
1
vote
0answers
23 views

Complex measures, Real Analysis Folland Problem 3.3.19

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 ...
0
votes
3answers
31 views

Generated sigma algebra and its countable subcollection [duplicate]

Let $\scr{C}$ be a collection of subsets. Prove that if $A \in \sigma(\scr{C})$ (sigma algebra generated by $\scr{C}$), then there exists a countable subcollection $\scr{C}_A$ of $\scr{C}$ such that ...
2
votes
1answer
22 views

Interchanging finite union of finite intersection of sets

I would like to exchange the set operation : $$\cup_{i=1}^m\cap_{j=1}^{n_i} A_{i,j}$$ to be $$\cap \cup A_{i,j}$$ but it is a bit confusing to keep track of the index, and deduce the formula. Compare ...
2
votes
1answer
44 views

about a product of random variables that converges weakly

Let $(\Omega,\mathcal{F},P)$ be a probability space. Suppose $f_n,g_n, n\in \mathbb{N}$ are sequences of functions on this space such that their product $f_ng_n$ converges weakly in $L^2$ to $h$, say. ...
1
vote
3answers
44 views

What can go wrong if we let sigma algebra to admit the union of uncountable union of elements?

By definition we only allow the union of countable infinite of elements to be also include the $\sigma$ field, why not uncountable many? Is there a historical view behind this?