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

learn more… | top users | synonyms (1)

3
votes
3answers
43 views

Show that continuous functions on $[0,1]$ satisfy this property

If $f \in C[0,1]$ prove that $$ \lim_{n \to \infty} n\int_0^1e^{-nx}f(x)dx $$ exists and find the limit. I can show that $|g_n|$ are bounded by $M=\max(f)$. After some test functions I suspect that ...
0
votes
0answers
11 views

How to prove the independent of these 2 $\sigma$-algebras?

Let $(\Omega,F,\mathbb{P})$ be a probability space. Let $(F_i)_{i\in\mathbb{N}^*}$ be a sequence of mutually independent sub-$\sigma$-algebras of $F$. I want to prove that: $$A=\sigma(\bigcup_{m\leq ...
3
votes
3answers
49 views

Prove $b-a \le \sum^n_{i=1}(b_i-a_i)$ by induction

Show that if the closed interval $[a,b]$ is covered by finitely many open intervals $(a_1,b_1), ...,(a_n,b_n)$, then $$b-a \le \sum^n_{i=1}(b_i-a_i)$$. I know that $(a_1,b_1), ...,(a_n,b_n)$ form an ...
0
votes
3answers
41 views

Prove that half-open sets in $\mathbb{R}$ are measurable

Self-learning these concepts, so please be tolerant with imprecise terminology... Defining the standard topology on the real line $\mathbb {R}$ as all the open intervals, a Borel $\sigma$-algebra is ...
1
vote
2answers
36 views

Lebesgue Integrability of $\left(\frac{1}{x}\right) \sin\left(\frac{1}{x}\right)$

Given $f:\mathbb{R}\rightarrow \mathbb{R}$ defined by $f(0)=0$ and $f(x)=\left(\frac{1}{x}\right) \sin\left(\frac{1}{x}\right)$ for $x\in \mathbb{R}-\{0\}$, can someone please give me a rigorous proof ...
0
votes
0answers
13 views

How to recover a measure from the product

Let $(X,\mathcal{F}_X), (Y,\mathcal{F}_Y)$ be measurable spaces and $\mu :\mathcal{F}\rightarrow [0,\infty]$ be a measure (assume that $\mathcal{F}\supseteq \mathcal{F}_X\otimes\mathcal{F}_Y$). I do ...
3
votes
1answer
39 views

$f\in L^{1}[0,1]$ Show $\lim_{n\to\infty}\int_{0}^{1}|f(x)|^{\frac{1}{n}}dx = m(\left\{ {x:f(x)\neq 0}\right\} )$

The following is from a Sample Exam question I am studying from, and the question has stumped me. $$f\in L^{1}[0,1]$$ $$\lim_{n\to\infty}\int_{0}^{1}|f(x)|^{\frac{1}{n}}dx = m(\left\{ {x:f(x)\neq ...
0
votes
1answer
20 views

If $A \cup B$ is measurable and $m(A \cup B) = m^*(A) + m^*(B) < \infty$ then A and B are measurables.

That's it. I've only been able to find that, since $A \cup B$ is measurable: $$m^*(A) + m^*(B) = m(A \cup B) = m_*(A \cup B) \geqslant m_*(A) + m_*(B)$$ Maybe using too that if C is measurable and D ...
0
votes
0answers
18 views

Measure of $|g| = ||g||_\infty$, with $g \in L_\infty $.

I would like to either prove or disprove that the measure of $|g| = ||g||_\infty$, with $g \in L_\infty $, is greater than zero. I'm thinking that I should use the definition of $||g||_\infty$= ...
1
vote
0answers
36 views

Show that minimum and maximum are contained in $V(\mathcal{R})$/ Stones' axiom

Let $\mathcal{R}\subset\mathcal{P}(\Omega)$ be a ring for some set $\Omega$. Consider $$ V:=V(\mathcal{R}):=\left\{\sum_{i=1}^n\alpha_i1_{A_i}: \alpha_i\in\mathbb{R}, A_i\in\mathcal{R}, ...
0
votes
0answers
11 views

I cast doubt on the theorem about function's measurability of semi-continuous function.

Def) $f$ is measurable if, for all $a$ in $\mathbb{R^1}$, $$\left\{\mathbf{x}:f(\mathbf{x})\gt a\right\} \text{ is measurable}~.$$ Def) $f$ is said to be upper-semi continuous if ...
0
votes
1answer
41 views

Prove a.s. convergence of random variables.

I need to prove this: Assume that you have a probability space $(\Omega, \mathcal{F},P)$, $X_t$ is a stochastic process which is jointly measurable with respect to $\mathcal{B}(\mathbb{R})\times ...
0
votes
0answers
20 views

what is the relation between X and ω

From the definition of random variable: In the special case of probability space (Ω, F, P), we use the phrase random variable (RV) to mean a measurable function, that is, X : Ω → R is a random ...
1
vote
1answer
31 views

Results on “subtraction” of measures and outer measures?

Most results I have seen involves addition of measures For example, let $m^*$ and $m$ be Lebesgue outer measure and Lebesgue measure respectively, then given $A = \bigcup\limits_{n = 1}^\infty E_n, ...
0
votes
1answer
18 views

I do not understand in a process of proving that $|H-E|=|Z|=0$ iff $|E=H-Z|$?

Notation $|E|_e$ is the outer measure of $E$. $|E|$ is the measure of $E$. A type $G_\delta$ means countable intersection of open sets. The theorem is $$ E \text{ is measurable if and only if } ...
2
votes
1answer
34 views

determining if a tail event

I am to determine if $$\{\sup X_n < \infty \}$$ is a tail event, the solutions are as follows: I don't understand how they got the line of equalities, specifically the last one, and why it holds ...
1
vote
0answers
11 views

Region under the graph of an unsigned measurable function is measurable

This was taken out of Tao's book on Measure theory: Let $f : \mathbb{R}^d \rightarrow [0, + \infty]$ be an unsigned measurable function. Show that the region $ \{ (x,t) \in \mathbb{R}^d \times ...
0
votes
0answers
18 views

Lusin theorem in measurable set $\Omega$ of $\mathbb{R}^N$

I have a question about Lusin theorem : We both know that Lusin theorem proof in X is locally compact Hausdorff space . My question is : "Can we change X into $\Omega$ ? " with $\Omega$ is ...
0
votes
1answer
14 views

Proof of a variation of the monotone convergence theorem where fn<f but fn isn't necessarily increasing.

That's basically all of it. $f_n$ and $f$ are all measurable and non-negative, $f_n\to f$ and $f_n\le f$, i want to prove that $\int_Rf=lim{\int_Rf_n}$ for $n\to \infty$ (Lebesgue integral). I know ...
0
votes
0answers
24 views

Limit of integrals is zero

Let $\lambda$ be a lebesgue integral on $[0,1)$. Define the intervals $I_{n,i}=\left(\frac{2i}{2n}, \frac{2i+1}{2n}\right)$ and $J_{n,i}=\left(\frac{2i+1}{2n}, \frac{2i+2}{2n}\right)$ for $0\leq i\leq ...
0
votes
1answer
19 views

Prove $\int_{\mathcal{Q_k}^c} \vert f \vert \to_k 0$.

Let $\mathcal{Q_k}=[-k,k]^n\subset \mathbb{R^n}$ for all $k\in\mathbb{N}$, the n-dimensional cubes, and $f$ any integrable (lebesgue) function. Prove that $\int_{\mathcal{Q_k}^c} \vert f \vert \to_k ...
2
votes
0answers
42 views

Continuous strictly increasing function with derivative infinity at a measure 0 set

Let $E\subset [0,1]$ with $\mu(E)=0$. Does there exist a continuous, strictly increasing function $f$ on $[0,1]$ so that $f'(x)=\infty$ for all $x\in E$ (in Lebesgue sense)? I think there exist such ...
1
vote
1answer
21 views

Convergence on $L_p$ spaces

I am trying to justify a simple result on convergence over $L_p$ spaces. The lemma is the following: Let $1\leq p<\infty$ and $0\leq f_k\nearrow f$ be measurable functions. Then $f_k\rightarrow ...
0
votes
1answer
19 views

$C(A)$ is $\|\cdot\|_2$-dense in $\ell_2(A)$

Let $A \neq \varnothing$ and $\cal {F}$$(A) = \{F \subset A \mid F$ is finite$\}$. Define $\ell_2 (A) =L^2(A, 2^A, \mu_C)$, with $\mu_C$ the counting measure. Let $C(A) = \{f: A \to \Bbb C, \exists ...
0
votes
1answer
29 views

Decreasing sequence of non-negative Lebesgue measurable functions and MCT

I'm learning about measure theory, specifically the Lebesgue integral of nonnegative functions, and need help with the following problem: Suppose that $f$ and $f_n$ are nonnegative measurable ...
0
votes
1answer
19 views

What is the limit of the lebesgue integral of the function sequence fn=1/n

If $f_n=1/n$ then what is the value of the following limit (Lebesgue integral): $$\lim_{n\to\infty}\int_Rf_n$$ I basically want to prove that a generalisation of the monotone convergence theorem ...
0
votes
0answers
9 views

Relation between Karamata's and Hardy-Littlewood's inequalities

In the field of (elementary) classical inequalities one of the most famous tools is the majorization inequality due to Karamata [1] (also known as Hardy-Littlewood-Polya). In its integral version, it ...
3
votes
1answer
19 views

Poisson Process independent Wiener Process using singular measures

I was reading some stochastic calculus of Jump processes and saw the result that if $W_t$ is Brownian and $N_t$ is Poisson both adapted to the $W_t$'s natural filtration then these processes are ...
0
votes
0answers
14 views

Characterization of the Jordan decomposition of a real-valued function of bounded variation from Folland.

This is a characterization of the Jordan decomposition of $F$ from Folland's Real Analysis. However, I can't see how the characterization makes sense. Let $F\in BV$ be a real valued function and ...
1
vote
1answer
16 views

Proof Explaination: Show the set of measurable sets is closed under finite union

I have a proof of the above claim but I think there are some mistakes, I have highlighted them I hope someone could help figure out exactly what is wrong. Given $\omega$ an outer measure on set ...
1
vote
1answer
37 views

What is the catch when introducing measure theory using $\sigma$-ring instead of $\sigma$-algebra?

I am currently using Matthew A Pons book Real Anaysis for Undergrad for introduction of measure theory In my opinion this book is unbelievably clear for almost all the sections EXCEPT the section ...
0
votes
1answer
72 views

Calculating a probability of a maximum event

Let $\{X_n\}$ be a sequence of IID random variabels with continuous distribution function. For each n let $E_n = \{X_n>X_i \text{ for all } i <n \}$ be the event that there is a record at time ...
4
votes
1answer
53 views

Differentiation under integral sign?

I am trying to understand the following argument given in a text book: Suppose $f \in L^1(\mathbb R^n)$, consider the function $\hat{f}(\zeta)= \int_{\mathbb R^n} \exp(-2\pi i X.\zeta)f(X)dX$. ...
0
votes
0answers
22 views

Intuition - Asymptotic Maximum Likelihood Estimator

Maximum Likelihood Estimation is quite clear to me when it is performed on finite sample sizes. The intuition of an obtained Maximum Likelihood estimate for given data $x_{1},...,x_{n}$, $n \in ...
0
votes
3answers
33 views

Problem with σ-Algebra definition

At the moment I'm doing my first steps in the probability theory but now I have some problems with the σ-Algebra. Here is the definition: $A \subseteq \mathcal{P}(\Omega)$ is called σ-Algebra if: $ ...
1
vote
1answer
18 views

Let $f=\sum_1^n a_j \chi_{E_j}$ be a simple function, compute $\int |f|^p=\int |\sum_1^n a_j \chi_{E_j}|^p$.

Let $f=\sum_1^n a_j \chi_{E_j}$ be a simple function where $\mu(E_j)\lt \infty$. I want to show that $f\in L^p$, but I don't know how to compute $\int |f|^p=\int |\sum_1^n a_j \chi_{E_j}|^p$. How ...
0
votes
0answers
12 views

Point-free notation for (not direct) sums of functions over products of spaces.

I am writing a paper, and there are lots of expressions containing integrals of form: $$\int_{X \times Y} \phi(x) + \psi(y) d\alpha(x,y) $$ where $\phi$,$\psi$ are abstract functions and $\alpha$ is ...
0
votes
1answer
19 views

Showing an inequality in the proof of $L^p$ is a Banach space for $1\le p \lt \infty$.

This is part of the proof that $L^p$ is a Banach space from Folland's Real Analysis, but there is a part that I don't understand. Suppose $\{f_k\}\subset L^p$ and let $G_n=\sum_1^n |f_k|$ and ...
1
vote
1answer
25 views

Pure states on $C(X)$ are exactly evaluations

Let $X$ be a compact Hausdorff space. I want to show that pure states are of the form $ \phi (f) =f(x)$. By Reisz Represenation Theorem states on $C(X)$ are of the form $\phi (f)= \int fd\mu$ where ...
1
vote
0answers
27 views

Background for understanding Measure Theory - preparation [closed]

I've recently come across a dilemma, which I hope I can get some opinion on. I am currently finishing my undergrad degree in mechanical engineering. I have a great interest in statistics, especially ...
1
vote
3answers
34 views

Verification of a proof in Measure Theory

Let $m$ be the Lebesgue measure on $\Bbb R$ and $f:\Bbb R\to [0,\infty)$ be a Lebesgue integrable function. Show that $\exists $ a measurable set $E\subset [0,\infty)$ such that $m(E)\neq m(f^{-1}(E)$ ...
0
votes
1answer
50 views

Uniform continuity with integral being finite

Let $f$ be a real valued uniformly continuous function on $\mathbb{R}$ that is lebesgue integrable. Show that $\lim_{|x|\rightarrow \infty}f(x)=0$. Suppose that ...
3
votes
2answers
38 views

Integrating over a sequence of sets $A_n$ with $\mu(A_n)\to0$

I am going through the proof of the following. Let $(X,\mu)$ be a measure space and $f\colon X\to\overline{\mathbb R}$ be a measurable function with finite integral. If $A_1,A_2,\dots$ are ...
0
votes
1answer
19 views

Equality in Lebesgue integration inequality

Let $(X,\mathcal{M})$ be a measure space and $\mu$ be a positive measure on it. Let $g:X\to \mathbb{C}$ be a complex valued $\mu-$integrable function. We know that $$ \left| \int_X g d\mu \right| \leq ...
0
votes
1answer
28 views

the measure of the set of x where f(x)= ∞ is zero

Let $(X,\boldsymbol M, \mu)$ be a measure space where the measure $\mu$ is positive. Let $f : X \to \mathbb R^+$ be a measurable function such that $\int f \text d \mu < \infty$. Let $N = \{x \in ...
-2
votes
1answer
69 views

why is $\{x\mid f(x)=\infty\}$ measurable? [closed]

why is $\{x\mid f(x)=\infty\}$ measurable? I know that $\{x\mid f(x)=a\}$ is measurable when a is finite. But textbooks rarely mention $\{x\mid f(x)=\infty\}$. It seems to be the opposite from the ...
1
vote
1answer
70 views

A problem related to Lebesgue integration.

I have following two problems: Suppose $$\int_E f \, dx = 0 $$ where $ f: R \to R$ is a measurable function that is strictly positive. Show that $E$ must be a null set. Next Suppose that $E$ is a ...
0
votes
1answer
81 views

How can one rigorously treat integration over jump discontinuities?

Suppose I wish to compute $$\lim_{n \rightarrow \infty} \int_{0}^{\frac{\pi}{4}} \tan ^n x \,dx$$ We can imagine that the tangent function goes to $0$, with the exception of the point $\displaystyle ...
5
votes
1answer
133 views

How does the cardinality of the set of all probability measure on a set $X$ change according to the cardinality of $X$?

I was wondering concerning the following problem: Take $X$ as a parameter space endowed with its Borel $\sigma$-algebra. What is the cardinality of $\Delta (X)$, understood as the set of all ...
0
votes
0answers
18 views

existence of $\mu$ such that $\int{p}d\mu = \sum_{k=1}^{n}p^{(k)}(k/n)$

If $n\geq 1$, show that there is a measure $\mu$ on [0,1] such that for every polynomial $p$ of degree at most n, $$\int{p}d\mu = \sum_{k=1}^{n}p^{(k)}(k/n)$$ I think we should prove the linear ...