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

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3
votes
0answers
36 views

Finding disjoint intervals from Cantor Set

Consider $C$ the classic Cantor ternary set in $[0,1]$. I am interested in the following problem: Find the largest constant $0<k<1$ such that it is true that any interval $[a,b] \subseteq ...
2
votes
1answer
32 views

Essential range of a function

Let $A_f$ be the set of all averages $\frac{1}{\mu(E)}\intop_{E}\,f\,d\mu$ where $E$ is of positive measure. What is the relationship between $A_f$ and $\mathbb{R}_f$? Is $A_f$ always closed? Are ...
2
votes
0answers
37 views

question about showing integral is zero

Given a real valued function $f:X\rightarrow\mathbb{R}$ we have that $(X,\mathcal{F},\mu)$ is a measure space and $f$ is measurable. If $\mu(\{x\in X\;|\; f(x)\in B\})=\mu(\{x\in X\;|\; -f(x)\in B\})$ ...
2
votes
0answers
24 views

Equality of two $\sigma$-algebras on $\mathbb{R}/\mathbb{Z}$

I need help with this problem: Let $p > 1$ a integer, $X = \mathbb{R} / \mathbb{Z}$, $\mathcal{B}$ the borel $\sigma$-algebra of $X$ and $T \colon x \mapsto px \text{ mod } 1$ I think that ...
3
votes
1answer
209 views

Why is it enough to prove the sentence?

I am looking at the proof of the theorem that for any rectangle the outer measure is equal to the volume. At the beginning of the proof there is the following sentence: It is enough to look at the ...
0
votes
0answers
29 views

About measurability for operator-valued functions

Being $E_1$ and $E_2$ Banach spaces, and working in a finite measure space, I have the following two definitions of measurability for a function $f:\Omega\to\mathcal{L}(E_1,E_2)$: $\bullet$ I say a ...
2
votes
1answer
51 views

A question on Lebesgue measure: Inequalities

Let $A_n$ be a decreasing sequence of Borel sets with finite but with measure $\geq \epsilon > 0$ for all $n$. Then there exists a compact set $B_n \subset A_n$ for all $n$ such that $\mu(A_n ...
1
vote
1answer
36 views

Sequence of functions divided by constants converge to zero

I am trying to show the following: If $\{ f_n \}$ is a sequence of a.e. real-valued measurable functions in X, and the measure $\mu(X) < \infty$, there exist positive constants $a_n$ such that ...
1
vote
1answer
22 views

Does integrating out a variable in a two-variable measurable function produce a measurable function?

This problem is not a mere consequence of Fubini’s Theorem, so I thought that it would be suitable for posting here on MSE. Let $ (X,\Sigma,\mu) $ and $ (Y,\text{T},\nu) $ denote $ \sigma $-finite ...
-2
votes
2answers
53 views

Complex Measures: Integrability

Approaches A complex measure decomposes into: $$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$ This gives rise to integrability as: $$f\in L(\mu)\iff f\in ...
3
votes
1answer
33 views

Characterization of the Haar measure in terms of the integrals of characters

I was reading a paper and I think that they used the following theorem: Let $G$ compact group and $\mu$ a probability measure on $G$. If $$\hat{\mu}(\xi)= \int_G \overline{\xi(x)} d\mu(x) = ...
0
votes
1answer
40 views

Lebesgue Mean Value Theorem

Disclaimer: This proof is taken out from Rudin, Real and Complex Analysis. Let $\Omega$ be a finite measure space $\lambda(\Omega)<\infty$. Denote the mean value by: ...
0
votes
1answer
26 views

Complex Functions: Integrability

Let $\Omega$ be a measure space with measure $\lambda$. Denote the space of simple functions by: ...
1
vote
1answer
34 views

Radon-Nikodym: Integrability?

Let $\lambda:\Sigma\to\mathbb{R}_+$ and $\kappa:\Sigma\to\mathbb{R}_+$ be finite measures on $\Omega$. Then by Radon-Nikodym: $$\kappa(E)\leq L\cdot\lambda(E)\quad(\forall ...
3
votes
1answer
50 views

Is the following a semiring?

I have the following problem: Let $f: X' \rightarrow X$ be any map and $\mathcal{H} \subseteq \mathcal{P}(X)$ a semring. Is $f^{-1}(\mathcal{H})$ a semiring? Thanks for your help!
1
vote
0answers
31 views

Property of uniformly tight random variables?

I'm stumped on the following question, which is problem 1.3.9 in the book Weak Convergence and Empirical Proceses by van der Vaart and Wellner. It is based on the following notion of asymptotic ...
1
vote
0answers
34 views

Showing that set contains no intervals.

Hi, I'm trying to solve a problem: Here goes: First part is true because sets $B$, $B'$, and $E$ are countable and hence $F$ is countable so it's Lebesgue measure is $0$, thus it contains no ...
0
votes
1answer
15 views

Bounds of integral in Power function

Here is the question: Let $X_1,X_2$ be iid uniform $(\theta,\theta+1)$. For testing $H_0:\theta=0$ versus $H_1: \theta>0$, we have two competing tests: $\hspace{15mm}\phi_1(X_1):$Reject $H_0$ if ...
0
votes
1answer
53 views

The measure of a rectangle is zero

Show that the measure of a rectangle is zero if and only if the inside is empty. Could you give me some hints what I could do??
1
vote
2answers
81 views

Abstracted Metric and Measure Spaces

As I am just beginning to study general topology and metric spaces in more and more detail, it seems to me that the metric space topology is entirely determined by the properties of $\Bbb R$, since ...
1
vote
0answers
36 views

Is this proof correct()?

Prove that for $p=\infty$ : $\|f\|_{L^{\infty}{(\Omega)}}=0 \implies f=0 $ a.e on $\Omega$ Proof $\|f\|_{L^{\infty}{(\Omega)}}=0 \implies ess$ $\sup |f| =0 \implies \inf\{a\in R| ...
1
vote
1answer
51 views

Suppose $A_n \supset A_{n+1}$ . Show $P(A_n) \searrow P(\cap_{n=1}^\infty A_n)$

I know the proof for if it is a monotonic increasing sequence of sets using the fact that $\sum_{i=1}^\infty P(B_i)$ goes to 0, where $B_i$ is a partition of ${A_i}$.
2
votes
1answer
16 views

A.e. equal functions for semifinite measure

As a homework in our Measure + Integration Theory course we had the following statement: If $(X, \mathcal{M}, \mu)$ is a $\sigma$-finite measure space and $f, g: X \to [0, \infty)$ are measurable ...
0
votes
0answers
20 views

Outer measure of set of points where a given function is infinite

I'm reviewing for a measure theory midterm and I'm having difficulty with the following problem. I don't need it fully solved, maybe just a few hints, I'm having trouble getting started. Let $f: ...
0
votes
0answers
22 views

Can I use the triangle inequality to ensure a unique measure?

Let $\mathcal{C}$ be a finite set of objects, $\Delta\mathcal{C}$ the set of probability measures on $\mathcal{C}$, and $\mathscr{U}$ be a finite set of linear functions, $u: ...
1
vote
0answers
25 views

Linearity of the integral without $\sigma$-additive measures

I was wondering how you could prove the linearity of the integral without using that measures are $\sigma$-additive. I have no clue of where to start, but let me state my question more precisely. ...
0
votes
0answers
25 views

Extension of measure beyond Jordan-measurable sets

I know that if a set $A$ is Jordan-measurable (according to the definition that can be found here in problem 8) with respect to measure $\mu$, then, for any measure $\tilde{\mu}$ that is an extension ...
1
vote
0answers
24 views

Extension of $\sigma$-additive measure beyond Lebesgue-measurable sets.

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа an unproven statement saying that the system of sets of $\sigma$-uniqueness for a $\sigma$-additive measure $m$ defined ...
0
votes
1answer
40 views

Smallest $\sigma$-algebra and $\sigma$-algebra generated by a function

I'm reading through the following theorem: Let $X=\{X_t,t\in T\}$ be a stochastic process. Then $\sigma (X)=\sigma ( \cup_{t\in T} \sigma (X_t))$ From my basic knowledge of measure theory, I ...
1
vote
1answer
16 views

Expectation of Truncated Random Variables (2)

This is a follow-up question from here. Let $X_n$ be a sequence of $iid$ random variables with zero mean, finite variance $\sigma^2$ and partial sum $S_n:=\sum_{k=1}^n X_k$. Let $$0<\delta<0.5$$ ...
4
votes
4answers
76 views

What are non-obvious examples of measures obtained from linear functionals by the Riesz representation theorem?

In chapter two of Rudin's "Real and Complex Analysis" there is a "Riesz Representation Theorem" that dominates the chapter. My understanding of the statement of the thm. is that given a complex-valued ...
1
vote
1answer
31 views

proof of DCT with weak condition(almost everywhere)

I have a question about a proof of the dominating convergence theorem, with weak requirements. Before I show the proof from the book, note that in my book you are allowed to integrate functions that ...
0
votes
0answers
23 views

Equality of extensions of Jordan measure

I find the following theorem in Kolmogorov-Fomin's Elements of the Theory of Functions and Functional Analysis (p. 280 of this Russian ed., p. 26 of 1963 Graylock English ed.): In order that two ...
2
votes
1answer
33 views

Understanding the measurability of conditional expectations

My question is about the conditional expectation of random variables with respect to a $\sigma$-algebra. I am having trouble getting an intuition behind the definitions among other things. I know ...
0
votes
1answer
26 views

dense subspace of $L^2(\Omega\times(0,T))$

I am trying to prove that the functions $f(\omega,t)=g(\omega)h(t)$ where $g\in G,\: h\in H,$ are dense in $L^2(\Omega\times(0,T))$ if $G$ is dense in $L^2(\Omega)$ and $H$ is dense in $L^2((0,T))$. ...
0
votes
0answers
17 views

Visualizing open covering of B contained in open covering of A (Outer measure)

Say a set A is contained in a B. We defined in class the outer measure as the inf of the sum of the open coverings because it's an overestimation. I understand why the outer measure of B is greater ...
0
votes
1answer
33 views

Sigma additivity on semi-algebra

Suppose we have a semi algebra $\mathcal{A}=\{(a,b]: -\infty<a \le b < \infty \}$. How to show that measure defined on this semi-algebra as \begin{align} \mu((a,b])=F(b)-F(a) \end{align} is ...
2
votes
1answer
34 views

Tail sigma field generated by i.i.d. sums

The random variables $(X_n)$ are i.i.d. real valued and in $L^1$. Let $S_n := \sum\limits_{i=1}^n X_i$ and $$\mathscr{G_n} := \sigma(S_n,S_{n+1},...).$$ Clearly the sigma fields $\mathscr{G_n}$ are ...
2
votes
0answers
24 views

Intersection of Probability measures

Let $X$ be a metric space with Borel $\sigma$-algebra $B(X)$ and suppose that $\mathbb{P}_1$ and $\mathbb{P_2}$ are two probability measures on $(X,B(X))$. Question: Suppose $G$ is an open set ...
0
votes
0answers
17 views

How to measure the similarity or divergence of two distributions with different supports?

Suppose $X$ and $Y$ are two random variables with the distributions $F_X$ and $F_Y$ on the same support $\Theta$.Then KL divergence $D_{KL}(X||Y)$ is a way to measure the statistical distance between ...
0
votes
1answer
33 views

An extension of Dominated Convergence Theorem

Is it possible to extend the dominated convergence theorem without the restriction of the limit function be integrable? In precise words, is the next statement true? Let $(f_{n})$ be a succession of ...
5
votes
1answer
89 views

A Fundamental Theorem of Calculus

Here is a problem I have been working on recently: Let $f \colon[a,b] \to \mathbb{R}$ be continuous, differentiable on $[a,b]$ except at most for a countable number of points, and $f^{\prime}$ is ...
0
votes
0answers
31 views

An aplication of Fatou Lemma [duplicate]

Let $(f_{n}) $ be a succession of m-measurable functions and let f be an integrable function (with respect to the measure m) all in X. Prove that if, lim $ f_{n} = f $ almost everywhere in X, and if $ ...
2
votes
1answer
20 views

Motivation for Definition of Measurable Function

I'm having trouble understanding why a function is defined as "measurable" if the preimage of every measurable set is measurable. I see the parallel to the definition of continuity, and the latter ...
2
votes
1answer
16 views

Subset of $A$ of arbirary small measure

Let $\mu(A)>0$. Show that for arbitrary small $\epsilon>0$, there exists a subset $B$ of $A$ such that \begin{align} 0 < \mu(B) < \epsilon \end{align} Assume that $\mu$ is not an attomic ...
0
votes
0answers
38 views

$\sigma$-additivity of an abstract measure

I know and have been able to prove the following lemma: Let $X$ be a set and $\mathfrak{M}$ a $\delta$-ring of subsets of $X$. The set $A\subset X$ is defined as measurable with respect to ...
0
votes
1answer
22 views

Show that $\sigma(S(C))=\sigma(C)$.

Let $ S(C)$ be an algebra generated by $C$ and let $\sigma(S(C))$ be a sigma algebra generated by $ S(C)$. Show that $\sigma(S(C))=\sigma(C)$. I can show $\sigma(C) \subset \sigma(S(C))$ this is ...
0
votes
1answer
42 views

Uncountable and Unbounded set of measure 0?

In my Real Anaysis course, the instructor posed a question. He asked to either give an example or to explain why it is not possible. He asked about an uncountable and unbounded set of measure 0. My ...
2
votes
2answers
32 views

$\sigma$-algebra generate by uncountable collection of subsets.

here is my question: Let $\sigma(F)$ be a $\sigma$-algebra generated by $F$ where $F$ is uncountable collection of subsets of $\Omega$. Let $A \in\sigma(F)$ then there exists a countable ...
1
vote
0answers
27 views

Extending a pre-measure on an sigma-algebra?

Consider on $\Bbb{R}$ the family $\Sigma $ of all Borel sets which are symmetric w.r.t. the origin, which is a $\sigma $-algebra. Is it possible to extend a pre-measure $\mu $ on $\Sigma $ to a ...