1
vote
0answers
50 views

Another version of the Poincaré Recurrence Theorem (Proof)

The task is to prove the following version of Poincaré's Recurrence Theorem: Let $(X,\Sigma,\mu)$ be a finite measure space, $f\colon X\to X$ a measurable transformation that preserves the ...
1
vote
0answers
25 views

A question on Abstract measure spaces

Let $(X,M)$ be a measurable space then 1) if $\mu $ and $\lambda $ are measures in $M$ st $\mu \ge $ $\lambda $ then show that $m$ defined as $\mu= \lambda + m $ is a measure 2) Prove that if ...
1
vote
2answers
90 views

Prove that a function with a measurable graph is differentiable [closed]

Let $f:[a,b]\to(0,\infty)$ be continuous and let $Gf:=\{(x,y)|y=f(x)\}$ be the graph of $f$. Prove that $Gf$ is measurable only if $f$ is differentiable in $(a,b)$?
3
votes
1answer
115 views

When is $\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$

An elementary question on Riemann - Integration: Under what conditions on $f$ is the following true: $$\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$$ If $f$ is bounded in $[a,b]$, then this is ...
0
votes
0answers
25 views

Find Jordan decomposition constructed by function.

$$ F(x) =\left\{ \begin{array}{l l} 0 && x \leq 0\\ 1-x && 0 \leq x \leq 1\\ 2x-4 && 1 < x \leq 2 \\ 1 && 2 < x \end{array} \right. $$ ...
1
vote
0answers
33 views

On $\sigma$-algebra generated by sets

Given $\mathcal{S}$ a collection of subsets of $X$ and $A\subset X$. To show that $\sigma(\mathcal{S}\cap A)=\sigma(S)\cap A$, where for any collection of $\mathcal C$ of subsets of $X$, $\mathcal ...
1
vote
1answer
19 views

$\sigma$-algebra generated by a set

I want to show that if $X$ is an uncountable set then $\mathcal{S}=\{\{x\}:x\in X\}$ generates the $\sigma$-algebra $\mathcal{A}=\{A\subset X: A$ is countable or $X\setminus A$ is uncountable$\}$. I ...
2
votes
1answer
71 views

Spivak's “Calculus in Manifolds” problems

I have some troubles with this problems. Problem 1.18: If $A \subset [0,1]$ is the union of open intervals $(a_i,b_i)$ such that every rational number of $(0,1)$ is contained in $(a_i,b_i)$, for ...
1
vote
2answers
74 views

Measure Theory Question 5

Let $E$ be a measurable set with $m(E)<\infty$ and $f$ be a measurable function on $E$ that is finite almost everywhere on E. For each $\epsilon>0$, show that there is a measurable set $F$ ...
0
votes
2answers
32 views

Regarding measurable functions

Let $(X,\mathcal A)$ be a measurable space and let $f:X\to \mathbb R$ and $g:X\to \mathbb R$ be mesurable functions. Let $G$ be an open subset of $\mathbb R^2$. We want to show that $\{x\in ...
0
votes
1answer
28 views

Problem regarding outer measure

Let $\mu^*$ be an outer measure on a set $X$. We want to show that $E\subset X$ is $\mu^*$-measurable iff for every $\epsilon>0$ there exists a $\mu^*$-measurable set $F\subset E$ such that ...
1
vote
2answers
33 views

Showing a set function to be measure

Let $(X,\mathcal A)$ be a measurable space and let $\mu:\mathcal A\to [0,+\infty]$ be a finitely additive set function such that $\mu(\emptyset)=0$. We want to prove that $\mu$ is a measure on ...
1
vote
1answer
47 views

Prove that a relatively compact subset of $L^p$ is bounded.

Let $p\in [1,\infty)$, $A\subset L^p(\mathbb R^m)$ relatively compact and $\lambda^m$ be the Lebesgue measure on $\mathbb R^m$. Prove: a) $A$ is bounded. b) $\lim_{y \to 0}\sup_{f \in A} ...
0
votes
2answers
44 views

How can I show that $L^{1}([0,1])$ is a separable metric space?

I am trying to show that $L^{1}([0,1])$ is separable. I know by definition that a space is separable if I can prove the existence of a countable and dense subset. However, I really dont know even how ...
2
votes
1answer
35 views

Lebesgue measurability of nearly identical sets

If $A$ is a Lebesgue measurable set such that $m^*(A\triangle B)=0$, then $B$ is Lebesgue measurable. I tried the problem as follows: $m^*(A\cup B)=m^*((A\triangle B)\cup(A\cap B))\leq ...
0
votes
1answer
40 views

Measure Theory question 4

If f is a non negative mesaurable function on $\mathbb R$ such that $\int f<\infty$, show that the set $\{x|f(x)>0\}$ can be written as a union of an ascending sequence of measurable sets of ...
1
vote
1answer
30 views

Identify the smallest sigma-algebra of subsets of $\mathbb{R}$ that contains the set [0, 1]

This is a past exam question which I've tried to do this myself, though I'm unsure of the solution. First of all, by the definition of a sigma algebra, I should include $\emptyset$. Then, ...
1
vote
0answers
29 views

Monotone convergence, measure-theory, is this excercise correct?

Here is the exercise: I have some questions: Is this correct when k starts with 1?, the Taylor series with e starts with 0? But does the zero disappear in some way?, I can not see how. I know that ...
3
votes
2answers
109 views

Excercise, measure theory

I need help with this excercise: This excercise is in a chapter where I learn the monotone convergence theorem and Fatous lemma, so I assume I shall use them. Since $\textbf{1}_Ef_n\rightarrow ...
2
votes
1answer
46 views

How to prove that a Lipschitz function is absolutely continuous?

$f:[a,b] \rightarrow \mathbb{R}$ is a Lipschitz function. How to prove that it is absolutely continuous on $[a,b]$? My attempt: Let $\epsilon> 0$. Set $d = \epsilon/M$. If $P = \{[x_i, y_i]\}$ is ...
2
votes
1answer
50 views

Prove that the boundary of a rectangle in $\mathbb{R}^n$ has measure zero

This is an exercise on my homework. The professor gave us a hint that we just need to prove that one dimension of the boundary has measure zero, and then we can take the countable union to have ...
0
votes
1answer
37 views

Show that the image of a zero measure set is of zero measure

I saw a topic on the subject but I did not quite understand, and it was a bit old and I didn't want to resurrect it. I am going in the right direction, I just need a little nudge. let $f: \mathbb ...
1
vote
0answers
16 views

How to show: $l^*(A)\le l^*(G_\epsilon)\le l^*(A)+\epsilon $ , $l^* $:outer measure

If $A$ s a Lebesgue measurable subset of $\mathbb{R}$ and $\epsilon\gt 0$ How to show: $\exists$ an open set $G_\epsilon \supset A$ such that $l^*(A)\le l^*(G_\epsilon)\le l^*(A)+\epsilon $, $l^* ...
1
vote
1answer
52 views

Finite subcover of pairwise disjoint open intervals

I have the following exercise: Prove that if $X$ is a countable compact subset of $ \mathbb{R}$, then for any $\varepsilon>0$ there is a finite collection of pairwise disjoint open intervals ...
0
votes
0answers
19 views

(L1* ∩ L2*) = (L1 ∩ L2)* for all languages L1 and L2 over the alpabet Σ={A,B} Is it true or false and why?

plz answer me Determine whether each of the following statements is true or false. If a statement is false, give a counterexample..... 1- $(L_{1}^{*} \cap L_{2}^{*}) = (L_{1} \cap L_{2})^{*}$ for ...
0
votes
0answers
12 views

Minkowski Content-2

Here is a link to the previous question about minkowski content: Minkowski Content My new question is as follows: do $n$-dimensional manifolds have $n$ dimensional Minkowski Content? For example: ...
1
vote
0answers
25 views

If $\nu$ is a complex measure, then $L^1(\nu) = L^1(|\nu|)$

I am trying to prove the following statement from Folland: If $\nu$ is a complex measure, then $L^1(\nu) = L^1(|\nu|)$ and if $f \in L^1(\nu)$, then $\left| \int f \; d \nu \right| \leq \int |f| \; d ...
4
votes
1answer
74 views

Minkowski sum of a positive Lebesgue measure set and $\mathbb{Q}$.

Let $A\subset \mathbb{R}$ be of positive Lebesgue measure, i.e. $\mu(A)>0$. Is it then true that $\mu(\mathbb{R}\setminus (A+\mathbb{Q})) = 0$? I am quite sure that if $\mu(A)>0$, then $A-A$ ...
1
vote
1answer
54 views

Integration of standard multivariate normal distribution

We should express the integral $I_{n}=\int_{\mathbb{R}^{n}}\exp\left(\frac{-\left\Vert x\right\Vert ^{2}}{2}\right)\mathrm{d}x$ using $I_1$. Where $\left\Vert x\right\Vert =\left(x_{1}^{2}+\cdots ...
0
votes
1answer
81 views

Show the outer measure of a union is the sum of the measures without Caratheodony

I am attempting the following question: Let $\mu^*$ denote an exterior measure, $\{A_j\}$ collection of disjoint, $\mu^*-measurable$ sets, show for any E: $\mu^*(E \cap (\cup(A_j)) = \sum ...
1
vote
3answers
58 views

Lebesgue measure problem

Let $f$ be a non-negative measurable function on $\mathbb{R}$, and suppose that $\int f=0$. Prove that the set where $f \neq 0$ is a zero set. The hint says to let $E_n=\{f>1/n\}$ and then compare ...
1
vote
1answer
158 views

Problem about $G_{\delta}$-set and $F_{\sigma}$-set

Prove if $E$ is any measurable subset of $\mathbb{R}$, then there are a $G_{\delta}$-set $G$ and a $F_{\sigma}$-set $H$ such that $H \subseteq E \subseteq G$, and such that $m(G$\ $H)=0$. In order to ...
0
votes
1answer
50 views

f being a Lebesgue integrable function on $(0, a)$ implies that $g(x) = \int_x^a (f(t)/t)dt$ is also integrable.

I need to prove: If f is Lebesgue integrable on $(0, a)$ and $g(x) = \int_x^a (f(t)/t)dt$, then g is integrable on $(0, a)$. I know that since f is integrable on the interval $(0, a)$ I have ...
3
votes
2answers
40 views

lebesgue measurable problem

Let $E$ be a Lebesgue measurable subset of $[0,1]$, and suppose that $m(E)>3/4$. Prove that $(-1/2,1/2) \subseteq E-E$. We use $E-E$ to denote the set $\{x-y:x,y \subseteq E\}$
0
votes
1answer
42 views

Measure Theory Question 3

Let $E$ be a measurable set with $m(E)<\infty$. Show that there is a descending sequence of open sets $\{G_n\}$ so that $E\subseteq G_n$ for all $n \ \epsilon \ \mathbb N$ and $ \lim_{n\to\infty} ...
4
votes
1answer
49 views

Show that $g\in\mathcal{L}^q(\mu)$.

Let $(X,\mathcal{A},\mu$) be a finite measure space and $p,q\in(0,\infty)$ such that $1/p+1/q=1$. Let $g\in\mathcal{M}(\mathcal{A})$ measurable function such that $$\int |fg|d\mu\leq C\|f\|_p$$ for ...
2
votes
1answer
33 views

Finding σ-Algebra

Let $\Omega=[0,1]$ and $Y(w)=\begin{cases}1, & \text{if $w\in [0,1/3]$} \\2, & \text{if $w\in (1/3,1]$} \end{cases}$ What is the $\sigma$-Algebra created by $Y$, $\sigma(Y)$? I am kinda ...
0
votes
1answer
52 views

Caratheodory's theorem and outer measure

I'm trying to show that $$\lambda(A)=\lambda(A\cap E)+\lambda(A\cap E^c)$$ where $\lambda$ is an outer measure, $A\subset \mathbb{R}$, $E \subset \mathbb{R}$, and $E$ is an elementary set; that is, ...
3
votes
1answer
39 views

A Problem in Convergence of Sequences of Random Variables

Let $\left( X_n \right)$ be a sequence of independent random variables on the measure space $(\Omega, \xi,\mathbb{P})$ with $$ \mathbb{P} \left( X_n=1 \right)= p_n \text{ and } \ \mathbb{P} \left( ...
0
votes
1answer
22 views

Question about an outer measure of a real subset

Let B be a subset of the real numbers, and let $$\mu$$ be an outer measure, and let A be the union of a finite number of real intervals. I have to show that $$\mu(B)=\mu(B\cup A)+\mu(B\cap A^c)$$ ...
3
votes
1answer
43 views

A problem on absolute continuity of measures.

Suppose $(X, \mathcal{F}, \mu)$ is a finite measure space, $\{\nu_n\}_{n\in\mathbb{N}}$ is a sequence of finite measures on $\mathcal{F}$ s.t. $\nu_n \ll \mu$ for all $n\in\mathbb{N}$ and ...
1
vote
1answer
34 views

Expectations of martingales

Consider a martingale $(M_n)_{n \geq 0}$ adapted to a filtration $(\mathcal F)_{n \geq 0}$ on a probability space $(\Omega, \mathcal F, P)$. Prove that, for each $k \leq n$; $$E(M_n M_k) = E(M_k^2)$$ ...
0
votes
0answers
34 views

An outer meassure not being probability measure

I have to prove that an outer measure is not necessary a probability measure. I have this example: Let $\Omega $ be infinity, for every $A \subset P(\Omega)$ $$ \mu^{*}(A) = \left\{ \begin{array}{l ...
1
vote
1answer
40 views

Symmetrisation of function

Consider the probability space $\Omega = \{-1, 0, 1\}$ with the $\sigma$-algebra of all possible events and a probability measure $P$. Consider also the smaller $\sigma$-algebra $$F = \{\emptyset, ...
1
vote
1answer
46 views

The behavior of Fourier transform near the origin

I'm attacking a homework problem, which I have reduced to the following: Let Schwartz function $f \in \mathcal{S}^1(\mathbb{R})$ be nonnegative and $\|f\|_{L^1} = 1$. Assume further that ...
1
vote
2answers
49 views

Triangle inequality for L2 norm

I am posed with a problem as below. Let $X$ be the collection of real valued functions that are measurable on $[0,1]$. For a given number $p\geq 1$ define the $L^p$ norm by ...
1
vote
1answer
36 views

How to determine $\sigma$-Algebra?

Let $\Omega$=$\Bbb R$ and $\mathcal R$ = {$A \subseteq \Omega : A \cap \Bbb R_+ $is Borel-set and $A \cap \Bbb R_- \in${$\emptyset, \Bbb R_-$} } What is the $\sigma$-Algebra generated by $\mathcal ...
2
votes
1answer
21 views

Defining $\sigma$-algebra on a subset

This is problem 2.6 from "Probability Essentials" by Jacod. Here's the question and my proof: Let $(\Omega, \mathbf{A})$ be a $\sigma$-algebra and let $B\in \mathbf{A}$. Show that $\mathbf{B} = ...
0
votes
0answers
17 views

Question about supremum involving non-constant function a.e.

We have a measurable function $f: X \to \mathbb{R}$. We assume that $f$ is non constant a.e. This means that $f(x) != c$ a.e. and also implies that $\mu*(\{x \in X : f(x) = c\}) = 0$. Also we assume ...
0
votes
1answer
51 views

Is this a $\sigma$-algebra(closed under contable union)?

Could I say that this $$ M=\{X\subseteq\Omega=[0,1):x\in X\iff y\in X\} $$ is an $\sigma$-algebra? I don't see whether it is closed under countable union. x,y are two singetons of $\Omega$ For ...