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

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2
votes
2answers
172 views

Infinite product of probability measures is a premeasure

This is an exercise from Real Analysis by Stein and Shakarchi (Chapter 6, Exercise 15). Given infinitely many measure spaces $(X_i, \mathcal M_i, m_i)$, each of which has measure 1, one can define an ...
1
vote
1answer
13 views

Locally compact metric space, Urysohn, approximation

Let $E$ be a locally compact separable metric space, $\mathcal{B}(E)$ be the $\sigma$-algebra of $E$ and $m$ be a $\sigma$-finite borel measure on $(E,\mathcal{B}(E))$. Assumtion There exists a ...
0
votes
2answers
24 views

Minkowski's Inequality in $L^\infty$ space

How can one show the inequality that $\|f+g\|_\infty ≤ \|f\|_\infty + \|g\|_\infty$? Can we use same real number $a$ for both $f$ and $g$ ? i.e, $$\|f\|_\infty = \text{ess} ...
7
votes
5answers
2k views

Meaning of measure zero

My book describes measure zero as following: A set of points on the $x$-axis is said to have measure zero if the sum of the lengths of intervals enclosing all the points can be made arbitrarily ...
1
vote
2answers
51 views

Continuous functions of minimal norm

Let $C$ denote the set of continuos functions on $[0,1]$ with the supremum norm. $M\subset C$ such that $$\displaystyle\int_{0}^{1/2}f(t)\, dt-\int_{1/2}^{1}f(t)\, dt=1,\; \forall f\in M$$ Show ...
-1
votes
0answers
11 views

Cinlar Ex. 1.15: Trace space of a measurable space.

In constructing the trace space on a subset of a measurable space, it seems one has to assume that the subset is an element of the original measure space's sigma algebra, i.e., measurable in the ...
1
vote
0answers
28 views

Hahn-Banach proof of existence of Haar measure

I'm reading these notes of Terry Tao on the Haar measure (and related topics) on a locally compact Hausdorff group $G$. When he goes through the construction of the Haar measure, he does so by way of ...
1
vote
1answer
13 views

prove that $F$ is $\mu\times\mathcal{L}$ measurable where $F(n,x)=\frac{(2n+1)^2\sin((2n+1)x)}{(n(n+1))^2}$

Let $\mu$ be the counting measure on $\mathbb{N}$ and $\mathcal{L}$ be the Lebesgue measure on $[0,\pi]$. Define the function $F$ on $\mathbb{N}\times\mathcal{L}$ by ...
0
votes
2answers
20 views

Looking for proof of theorem on complex measurable functions

In University I have been given the following result: If $f:X\to\mathbb{C}$ is a measurable function in $L^1(X,\mathcal{E},\mu)$ with $\mu$ being finite, and there exists a closed set ...
5
votes
3answers
32 views

Borel measurable functions $f:[0,1]\to[0,\infty)$ which cannot be expressed as pointwise limit of nondecreasing sequence of step functions

An interval in this problem may be open, closed or half open. We regard a singleton $\{a\}$ as being an interval also. A step function is a real valued function on $\mathbb{R}$ which is a linear ...
1
vote
2answers
54 views

Prove that a intergral over $\mathbb R$ is finite

Let $K\in \mathcal L_1(\mathbb R)$ and $f$ be measurable and bounded on $\mathbb R$ such that $\lim_{|x|\to \infty} f(x)=0$. Define $$F(x):= \int _{\mathbb R} K(x-s)f(s)\;ds \qquad (x\in \mathbb R)$$ ...
2
votes
1answer
36 views

Completely stumped on exercise cooncerning the characterisation of Jordan measure - would anyone be willing to give a hint?

In Terry Tao's notes on measure theory he has the following exercise, I have no idea how to deal with the last statement, I would really appreciate it if someone could give a hint for the final case. ...
0
votes
1answer
20 views

The smallest filtration for which a sequence of random variables is adapted

Let $X_1, ..., X_n$ be a sequence of random variables. Show that $\hspace{60pt}$ $\mathcal{F}_n$ = $\sigma(X_1, ..., X_n)$ is the smallest filtration such that the sequence $X_1, ..., X_n$ is ...
0
votes
1answer
24 views

Dimension of rectifiable curve

Suppose $\Gamma$ is a rectifiable curve (means a curve with finite length), I want to prove that the Hausdorff measure of the intersection of it with closed subset $A\subset \mathbb{R}$ is 0, i.e ...
2
votes
2answers
270 views

counting measure on naturals not equal to limit

Given a $\sigma$-algebra, $\mathcal{A}$, of all subsets of $\mathcal{N}$ with a counting measure $\mu$. Give a decreasing sequence $\{A_n\}$ of sets in the $\sigma$-algebra such that $\mu(\cap_k A_k) ...
1
vote
0answers
35 views

a question on quasi-invariant measures (with respect to the irrational rotations) on the unit circle

Fix a $\sigma$-finite atom-less measure $\mu$ on the unit circle, which is quasi-invariant and ergodic under the rotation $T$ of the angle $2\pi\theta$, $\theta$ irrational. By a well-known result of ...
3
votes
3answers
259 views

Random Variables that aren't measurable

I've been reading through a math. stats. book, and I'm a little confused with the concept of measurable random variables. The book states: Let $(E, \mathcal{E})$ and $(F,\mathcal{F})$ be two ...
1
vote
0answers
53 views

Every projection of the square of the middle thirds Cantor set contains an interval

Let $C_\lambda$ the cantor set which is defined by the IFS $\{\lambda x,\lambda x+(1-\lambda)\}$ and also let $E=C_\lambda\times C_\lambda$.Suppose $\lambda =\frac 1 3$, we get the standard ...
6
votes
1answer
155 views

Equivalence of the Lebesgue integral and the Henstock–Kurzweil integral on nonnegative real functions

Let $f:[a,b]\to[0,\infty)$ ($\mathbb{R}\ni a<b\in\mathbb{R}$), and fix $c\geq0$. I want to establish the equivalence of the concepts of Lebesgue integrability and Henstock–Kurzweil integrability ...
1
vote
1answer
13 views

A relation between the inner and outer jordan measures

I'm studying measure theory and I was thinking about the following question: Is it true that whenever $A\subset B\subset \mathbb{R}^n$ are bounded, $$m^*(B-A)=m^*(B)-m_*(A)?$$ I have easily ...
1
vote
0answers
31 views

Approximation of Conditional Expectation with Respect to “Y” Using Simple Approximation of “Y”

Background. (TL:DR you can skip to Question. below.) This is a followup question to one of my previous questions (linked here) on this website. In short, the other question was about how to express ...
2
votes
1answer
19 views

$L_1$ convergence of $\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$

Does the sequence $f_n=\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$ on $(0,1)$ converge in $L_1$? It converges to zero pointwise and I think it converges in $L_1$ as well since ...
3
votes
1answer
50 views

Bounding $P(X \le \tau)$

I am trying to upper bounding $P(X \le \tau)$ where $X$ is non-negative r.v. and where $\tau \le 1$. I have become aware of the Reverse Markov inequality that says that, if $P(|X|\le a)=1$ then for ...
2
votes
1answer
57 views

Relationship between measure theory and real analysis

Does measure theory generalize real analysis to abstract spaces? For example, you can now talk about convergence even on unordered fields.
1
vote
1answer
184 views

Characteristic function

Question: Let $X_1$ and $X_2$ denote independent real-valued random variables with distribution functions $F_1$, $F_2$, and characteristic functions $\varphi_1$, $\varphi_2$, respectively. Let Y ...
4
votes
1answer
54 views

What are some practical applications of measure theory apart from providing theoretically rigourous foundations?

It seems that measure theory has a very good theoretical purpose, in that it provides a rigorous framework to define a lot of what we do in analysis. However, I have a hard time thinking of a ...
1
vote
1answer
35 views

Measurable set limit

If $\forall n \in ℕ$ , $ f_n: (X,M) \rightarrow (\overline{\mathbb{R}},B) $ are measurable. (where X is any space, M is a sigma algebra, B is Borel sigma algebra) Prove that the set $A = \{x\in X: ...
6
votes
1answer
401 views

Application of Monotone class Theorem in the proof of Kunita-Watanabe Inequality

The Kunita-Watanabe Inequality says: Let $X,Y$ be two continuous locale martingales and $H,G$ two product-measurable functions on $(0,\infty)\times \Omega$, then $$ \int_0^t|G_s||H_s|d|\langle ...
0
votes
1answer
16 views

Infinite products of scaled indicator variables: almost sure convergence vs. uniform convergence of the sample mean

Let $\frac{X_i}{2}\sim Ber(0.5) \implies E[X_i]=1$, and let $Y_n=\prod\limits_{i=1}^n X_i$. Since the $X_i$ are iid, $E[Y_n]=1,\;\forall n<\infty$. However, something weird appears to be happening ...
2
votes
1answer
24 views

How to show that the function $g(x)=x|\sin(x^{-1/2})|$ is absolutely continuous?

I am having trouble showing the on $[0,1]$, $g(x):=x\mid\sin(x^{-1/2})\ \mid$ when $x>0$ and $0$ is $x=0$ is absolutely continuous. I was instructed to try: $\ m(A) < \delta \Rightarrow ...
16
votes
2answers
280 views

Measure - exercise 22 from Folland

I'm doing some exercises from Folland's real analysis book. Exercise 18 is done and should help to do exercise 22, but I'm stuck. The definition of completion is given below. This is not ...
0
votes
0answers
13 views

Automophism of G and Haar measure

Let $G$ be a locally compact group (written additively), $\lambda$ an automophism of $G$, and $\alpha$ a Haar measure in $G$. As the Haar measure is unique up to factor constant, $\lambda$ transform ...
1
vote
1answer
53 views

integral of lebesgue function is continuous

Let F be a lebesgue integrable function on $(0,\infty)$. For $0 \le t < \infty$, define $g(t)=\int_{0}^{\infty} e^{-tx}F(x)dx$. Can someone explain why $g$ and $g'$ are continuous over ...
2
votes
5answers
108 views

Is there a proof that $\int \frac {dx}{x}=\ln |x|+c$?

Is there a proof that $$\int \frac {dx}{x}= \ln|x|+c$$ for $x\neq 0$ I would be interest for any replies or any comment.
0
votes
1answer
71 views

A question regarding non-(Lebesgue)-measurable sets in models of ZFC+$2^{\aleph_0}$=$\aleph_2$

Let $\mathscr V$ represent a set of Vitali's type. It is known that $|\mathscr V|=2^{\aleph_0}$. Does $\mathscr V$ have any measure-theoretic properties in models of, say, ...
3
votes
1answer
29 views

Relation between the modulus of integrability and $L^p$ spaces

Let $(X,\mu)$ be a measure space with $\mu(X)<\infty$. Given an integrable function $f$ on $X$, we can quantify its integrability in multiple ways. One is the modulus of integrability, which is a ...
6
votes
0answers
51 views

Why does the union of all open null sets is itself a nullset for second countable space?

On the online Encyclopedia of mathematics, it is written "The existence of a countable base guarantees that the union of all open μ-null sets is itself a nullset." See: ...
1
vote
2answers
51 views

Probabilistic implications of the existence of non-measurable sets

Measure theory and probability theory are deeply connected through the interpretation of subset measures on the sample space as probabilities of events. A major (and somewhat disturbing) result from ...
1
vote
2answers
27 views

Using the MCT to evaluate the integral of a series

I'm studying for my Measure Theory final and I've come across a question that I can't seem to find an answer for. For each $n \in \mathbb{N}$ set $E_n:=[n,2n]$ and let $f:\mathbb{R} \to \mathbb{R}$ ...
1
vote
1answer
97 views

Fatou: Reverse?

Attention The usual problems are about absolute convergence: $$\int|g_n|\mathrm{d}\mu\quad(g_n=f_n,f-f_n,s_m-s_n,\ldots)$$ (There Fatou may help out!) But as proceeding with Fatou one encounters ...
3
votes
1answer
16 views

Independence of random variables involving Brownian motion

I am reading a book on stochastic analysis and I don't understand the following (i.e. don't know how to prove it rigorously): Let $B$ be a standard Brownian motion and $\{ \mathcal{F}_t \}$ be the ...
3
votes
3answers
77 views

Cardinality of non-Borel sets

Assume ZFC. Let $B\subseteq\mathbb R$ be a set that is not Borel-measurable. Clearly, $B$ must be uncountable, since countable sets are always Borel being a countable union of measurable singletons. ...
1
vote
1answer
54 views

Relying two points by an almost-geodesic omitting a singular set a.e.

I failed to give an appropriate title to the question, so any suggestion for a better title is welcome: Here's the question: I would like to prove the following result: Given $\varepsilon>0$, a ...
6
votes
0answers
62 views

Borel measurability is a local property

I am looking at Exercise 5.2 (page 44) in "Real Analysis for Graduate Students" by Richard Bass. Let $f:(0, 1)\to \mathbb{R}$ be a function such that for every $x\in (0, 1)$, there exist ...
4
votes
1answer
71 views

Convergence in measure implies convergence almost everywhere (on a countable set!)

Here is an interesting problem from "Real Analysis for Graduate Students" by Richard Bass (which is an amazing book, by the way). Suppose $(X, \mathcal{A}, \mu)$ is a measure space, and $X$ is a ...
1
vote
0answers
27 views

Existence of measure given the margin is a step function

Suppose $Q:[0,1]\to [0,1]$ is given by a nondecreasing step function $$Q(x)=A, if \phantom{0}0\leq x < x^*$$ $$\phantom{0000} = B, if\phantom{0} x^*\leq x\leq 1 $$ s.t. $$A,B\in[0,1] ...
1
vote
0answers
32 views

Measurability of the points of (strict) increase for Stochastic Process

Given a background space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ , I'm considering a stochastic process $X:=(X_{t})_{t\geq0}$ with distribution $X(\mathbb{P})$ on ...
4
votes
4answers
347 views

Is the outer measure of $A\cup B$ equal to the sum of their outer measures if $A\cap B=\varnothing$?

I understand that Lebesgue outer measure on $\mathbb R$ is not countably additive. But if there are two disjoint sets, does the outer measure of their union equal the sum of their outer measure? Can ...
4
votes
2answers
49 views

If $\int_E f=\int_E g$ then $f=g$ a.e.?

Is the converse of the following statement is true? Let $f$ and $g$ be two bounded measurable functions on a set $E$. If $f(x)=g(x)$ a.e. on $E$ then $$\int_E f=\int_E g$$ Here is my proof for ...
2
votes
1answer
25 views

A Pasting lemma for measurable functions

I have the following setting: Let $(\Omega,\Sigma)$, $(\Gamma, \mathcal{C})$, $(X_{1},\mathcal{B}_{1})$, and $(X_{2},\mathcal{B}_{2})$ be measurable spaces such that $\Omega = X_{1}\cup X_{2}$, ...