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

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4
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1answer
83 views

A function $f:\mathbb{R} \to \mathbb{R}$ with infinite norm but finite weak seminorm

Let $0<q\leq p<\infty$. For $f:\mathbb{R}\to \mathbb{R}$, we define the norm \begin{equation} \|f\|_{p,q}=\sup_{a\in \mathbb{R},r>0} r^{\frac{1}{p}} \left(\frac{1}{r} \int_{a-r}^{a+r} ...
0
votes
2answers
55 views

A function is a.e. equal to a polynomial.

Let $f\in{L^p}$. For $t>0$, let $P_{t,n}(x)$ be a collection of polynomials of degree less then or equal to $n$ in the variable $x$ and the family is given by $t$ such that ...
1
vote
1answer
61 views

Using Riemann integral to define Lebesgue Integral

In the text I'm working through, the Lebesgue integral is related to the Riemann integral as follows: For some non-negative, real valued function $f$ on $\Bbb{R}$, set $E_y=\{x:f(x)>y\}$ and ...
2
votes
0answers
96 views

Kolmogorov extension-type result

I would like to prove the following, using the standard Kolmogorov extension theorem (e.g. http://en.wikipedia.org/wiki/Kolmogorov_extension_theorem): Let $(\Omega, \mathscr{F}, P)$ denote our ...
3
votes
3answers
287 views

Two random variables from the same probability density function: how can they be different?

The definition of $X$ as a random variable according to Wiki is as follows: $Let (\Omega, \mathcal{F}, P)$ be a probability space and $(E, > \mathcal{E})$ a measurable space. Then an $(E, ...
1
vote
2answers
53 views

Approximation of measurable subsets of $\mathbb{R}^\mathbb{N}$

Let $\mu$ be a Borel measure and let $A$ be a $\mu$-measurable subset of $\mathbb{R}^\mathbb{N}$. I wish to approximate $A$ using a degenerate subset of $\mathbb{R}^\mathbb{N}$; more precisely, I wish ...
1
vote
1answer
44 views

Probability distribution “similar” to Gaussian.

Does there exist a distribution A other than Gaussian such that: 1) linear combination of random variable from A is distribution A 2) easy to integrate, for example find entropy Thank you
0
votes
1answer
73 views

weak-*-convergence of measures ==> convergence of the total mass?

Let $X = [0,1]$. Let $\mu_n$ be a sequence of regular signed Borel measures on $X$, which converges to a measure $\mu$ on $X$ in weak-star, i.e. for any $f\in C_0(X)$, we have $\int_X f \mu_n(dx) \to ...
2
votes
0answers
63 views

Weak convergence in $L^1$

Does anyone have a reference for the following statement or similar ones? Let $U$ be an open bounded set in $\mathbb R^n$ and let $f\in C^0(U\times S^1)$. Then the sequence $f_m (x):=f(x,mx_i)$ ...
0
votes
1answer
35 views

I have a question about point wise convergence concerning Lebesgue measurable sets.

If I have a sequence $\{E_n\}_{n=1}^\infty$ of $\mathcal{M}$-measurable sets, and $E=\cup_{n=1}^\infty E_n$. How can I show that $\chi_{E_n} f\rightarrow\chi_Ef$ pointwise?
2
votes
1answer
56 views

Subsets of Finite Lebesgue measure sets

have a measure theory question that's got me stumped. Let $A$ have finite, positive Lebesgue measure $p$. Show that for all $0<q<p$, there exists a subset $B$ of $A$ with measure $q$. I know ...
1
vote
1answer
44 views

Please explain this conditional expectation equality

I understand that E[X|Y] is a random variable. But I am kind of confused about the following : $$ \int_{\{Y=y_i\}} E[X|Y] dP = E[X|Y=y_i]P(Y=y_i) $$ In the above, P is a probability measure , then ...
1
vote
1answer
54 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
262 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 ...
0
votes
1answer
77 views

$f '$ is not Lebesgue integrable on $[-1,1]$

Let f be that function from R to R defined by f(x)= 0 if x=0 x^2 sin(1/x) if x not = 0 show that the function f' is ...
0
votes
1answer
23 views

Show DE is closed

Does anyone know how to deal with the following kind of question? Here is the definition of Dynkin System: Also definition of intersection-closed:
1
vote
1answer
45 views

Not use Lemma to prove the Borel field B(R)=σ({(a,b): -∞<a<b<∞}).

Prove Here, I have found some relative info: Definition of Borel field: Lemma: But if do NOT use Lemma, how can I prove the above Borel field B(R)=σ({(a,b):-∞< a< b < ∞})? Here are ...
1
vote
1answer
77 views

Rearranging a double series; what's the rigorous argument behind this?

Recall that a measure zero subset $A \subset \mathbb{R}$ is one such that for any $\epsilon > 0$ there exists a sequence of intervals $J_n$ whose union cover $A$ and such that $\sum_n l(J_n) < ...
3
votes
1answer
106 views

Filtrations and Sigma-Algebras

I have been practising a question set by my lecturer and try to verify the answer, unfortunately I am unable to understand the following question and answer. $\textbf{Question:}$ Let ...
0
votes
1answer
62 views

Question about Dynkin system

Let $\Omega \neq \varnothing$ and $D$ be a Dynkin System in $\Omega$. For all $E\in D$, show that $D(E)$ is closed under taking complements where: $$D(E): \left\{F\in P(\Omega): F\cap E\in ...
1
vote
1answer
49 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 ...
3
votes
1answer
111 views

$n^n$ are the moments of a measure on the non-negative real line?

I would like to know if the numbers $1,1,2^2,3^3,\dots, n^n,\dots$ are the moments with respect some measure $\mu$ on $[0,+\infty)$, i.e., if there exists such a measure $\mu$ with $$n^n=\int_0^\infty ...
0
votes
1answer
42 views

Alternate definition of Lebesgue integral

I'm am essentially reasking this question Two definitions of Lebesgue integration but for a different approach. The answer given is intuitive, however I wondering how exactly to show the equality by ...
0
votes
1answer
38 views

Silly question on Measures

Let $(X,S,\mu)$ be a measure space. we have (1) $\mu$ is said to have the property $P_1$ if there is a countable family $\mathcal{C}$ of compact subsets of $X$ such that for every $U\in S$ and ...
0
votes
3answers
726 views

Measure Theory and Functional analysis exercise book

I'm looking for a big collection of exercises of functional analysis and measure theory. I know a lot of theory books which present some excercises (Brezis, Rudin, Lang, Royden, and others) but I was ...
2
votes
0answers
99 views

Borel cantelli lemma application.

For each fixed $C>0$ write $$A_{c}=\{x\in [0,1]:\mid x-\frac{p}{q}\mid >\frac{c}{q^3} \text{for every relatively prime pair} (p,q)\in \mathbb{N}\}$$ Prove that each $A_{c}$ is measurable and ...
1
vote
1answer
74 views

A little help on properties Lebesgue integration.

Suppose $f$ is a nonnegative $\mathcal{M}-\text{measurable}$ function and $\{E\}_{n=1}^\infty\subset\mathcal{M}$ with $E_1\supset E_2 \supset \cdot \cdot \cdot $. Further suppose $\int_\mathbb{R}f \ d ...
0
votes
1answer
29 views

Prove measurability of a function defined from two measurable functions

You have two measurable functions $L$ and $U$ defined on $([0,1],\mathcal{B}[0,1],Leb)$. Define $$ f = \begin{cases} L & \text{if }L=U \\ 0 & \text{otherwise} \end{cases} $$ The text says ...
0
votes
1answer
34 views

Is $f$ integrable in $L(X,\mathcal{X},\mu)$

Is $f$ integrable $L(X,\mathcal{X},\mu)$ $\mu(E)=\sum_{n\in E\cap\mathbb{N}} |n^2+n-6|$ $f:\mathbb{R}\rightarrow \mathbb{R_+}\cup\infty$ $f=(x-2)^{-4}$
3
votes
1answer
94 views

Topology of weak convergence

Edited: Thanks to etienne. I start with a compact metric space $(X,d)$. Then I consider the collection of finite measure $\mathcal{M}$ on $X$ and I equip $\mathcal{M}$ with the topology of weak ...
0
votes
1answer
57 views

Measure of a Particular measurable set

I am trying to find two measurable sets A and B such that $\mu (A+B)>0$ but $\mu (A)=0$ and also $\mu (B)=0$. The text book hint says that consider A=$C$ and B=$C/2$. But I do not understand how to ...
2
votes
3answers
137 views

help with showing completeness

Let $\left\{H_n\right\}_{n=1}^\infty$ be a sequence of Hilbert spaces and let $H=\left\{\left\{x_n\right\}:x_n\in H_n, \sum ||x_n||^2<\infty \right\}$. Define the inner product as ...
2
votes
1answer
38 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} = ...
1
vote
1answer
27 views

Why is this set an event?

As a part of a setup for another problem, my text remarks that it can be used without a proof that if $X_1, X_2, \ldots$ are random variables then $$C:=\{ \omega\in\Omega \ | \ \sum X_n(\omega) \ ...
0
votes
1answer
26 views

The supremum of numbers $c$ such that $f\ge c$ a.e. is finite

We have a measurable function $f: X \to \mathbb{R}$. We assume that $f$ is non constant a.e. This means that $f(x) \ne c$ a.e. and also implies that $\mu^*(\{x \in X : f(x) = c\}) = 0$. Also we ...
0
votes
2answers
43 views

Showing a measurable function is zero a.e.

Let $f:(a,b)\rightarrow{\mathbb{R}}$ be a non-negative measurable function such that $\int_{a}^{b}{f}dx=0$. Then $f=0$ a.e. To prove this result, let $A:=\{x\in{(a,b)}:f(x)>0\}$. Then on ...
1
vote
0answers
44 views

Does Lebesgue measure $ f $ is 0?

suppose $ f:[a,b]\rightarrow [c,d] $ be a function of one-to-one and onto and continuous and $ m$ be Lebesgue measure on $ [a,b] $. Do the function $f$ image null subset of $[a,b]$ to the null set?
0
votes
1answer
64 views

Continuity of probability measure

Sorry, I just wanted to know whether I understand this correct. Let $(x_n)$ be an increasing sequence such that $x_n \rightarrow a$, then we have for the probability measure on an arbitrary ...
0
votes
1answer
66 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 ...
1
vote
1answer
55 views

Is P finitely additive when P(A) = 0 if A is Finite?

Would the answer to the exercise below be "No"? As you would have P(A∪B) = P(B) because P(A) = 0 if A is finite. Suppose that Ω = N is the set of positive integers, and P is defined for all A ⊆ Ω by ...
1
vote
0answers
46 views

Inverse map measurable

We said that a function $f:X \rightarrow \mathbb{R}$ is measurable iff we have that for all $I_a:=(a,\infty)$, $a \in \mathbb{R}$ $f^{-1}(a,\infty)$ is measurable. Now I want to show that ...
0
votes
1answer
122 views

Decomposing real line as a union of a nullset and a set of first category

$\Bbb R$ can be written of the form $A\cup B$ such that $A$ is of measure zero and $B$ is of the first category! can anybody prove this? I guess $A$ must be an $G_{\delta}$ set which is dense in $\Bbb ...
0
votes
0answers
84 views

Slight issue with Lebesgue Integration (Dominated Convergence Theorem)

I have the following question: Prove that $\displaystyle\lim_{n \rightarrow \infty} \int_0^{n^2} e^{-x^2}n\sin\left(\frac{x}{n}\right) \ dx = \frac{1}{2}$ When doing this question, I showed ...
2
votes
0answers
36 views

Existence of invariant Borel measures in infinite-dimensional separable Banach spaces.

Let $\mu$ be a Borel measure in an infinite-dimensional separable Banach space $X$. We say that $h \in B$ is admissible(in the sense of quasiinvariance) for the measure $\mu$ if measures $\mu$ ...
2
votes
1answer
59 views

Below bound of the mesure of a finite intersection

Let $(X, \mathcal{M}, \mu)$ be a measure space, with $\mu(X)=1$. If $A_{1}, A_{2}, ..., A_{n} \in \mathcal{M}$, prove that $$\mu \left(\bigcap_{j=1}^{n} A_{j} \right) \geq \sum_{j=1}^{n} \mu{(A_{j})} ...
3
votes
0answers
80 views

Show that limit of a sequence of finite measure is a finite measure

Suppose $\mu$ is a finite measure on a $\sigma$-algebra $\mathcal{F}$ of a set $X$, $\{\nu_n\}_{n\in\mathbb{N}}$ is a sequence of finite measures on $\mathcal{F}$ s.t. $\nu_n$ is absolutely comtinuous ...
0
votes
1answer
43 views

There exists a measure such that the sum of derivatives is the integral

This is a homework question in functional analysis. If $n \geq 1$, show that there is a measure $\mu$ on $[0,1]$ such that $\displaystyle\sum_{k=1}^n p^{(k)} \left( \dfrac{k}{n} \right) = \int p ...
8
votes
4answers
218 views

Does the operator $T(f)(t) := f(t) - f(0)$ preserve measurability?

Denote by $\mathcal{B}$ the Borel field on $\mathbb{R}$, denote by $\mathbf{C}_{\left[0,\infty\right)}$ the set of continuous, real-valued functions over the domain $\left[0,\infty\right)$ and denote ...
0
votes
1answer
42 views

convergence ( in measure theory)

Does $g_n=n \mathcal{X_{[1/n,2/n]}}$ converge to g=0. Leb. measure My idea is since the $[1/n,2/n]$ goes to zero as n goes to infinity, there is no element in the interval, so the charateristic ...
0
votes
2answers
63 views

Lebesgue measure identity

Let $A,B\subset \Bbb R$ non-empty. Let $E,F\subset \Bbb R$ measurable such that $A\subset E ,B\subset F$ and $m(E\cap F)=0$. Then $m^{*}(A\cup B)=m^{*}(A)+m^{*}(B)$. I need help with the proof of the ...