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

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5
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0answers
25 views

What is the integral of $e^{a \cdot x+b \cdot y}$ evaluated over the Koch Curve

What is $$\int_{K} e^{a \cdot x+ b \cdot y} \mu(x,y)$$ where $K$ is the Koch curve and $\mu(x,y)$ is a uniform measure. Attempt: I can evaluate the integral numerically and I have derived a method ...
0
votes
1answer
12 views

Prove that the image of a curve has zero content

Definition: A set $A \subset \mathbb{R}^2$ is said to have zero content if, for all given $\varepsilon >0$, exists a finite collection of rectangles $A_1, \dots, A_n$ such that $A \subset ...
3
votes
2answers
52 views

The norm $\|f_n-f\|_{L^1} \to 0$ but $f_n \not\to f$

A classmate and I are studying this following question from Stein-Shakarchi, Chapter 2, Exercise 12: Show that there are $f \in L^1(\mathbb{R}^d)$ and a sequence $\{f_n\}$ with $f_n \in ...
2
votes
1answer
32 views

If E is measurable, then $\delta E$ is measurable.

Problem: If $\delta =(\delta_1,\delta_2,\cdots,\delta_d)$ is a d-tuple of positive numbers $\delta_i>0$, and $E$ is a subset of $\mathbf{R^d}$, we define $\delta E$ by $\delta E = ...
1
vote
0answers
14 views

Characterisation of continuous functions whose support is compact

I saw this question some time ago but I can't remember where. Basically the question ask to find the set of all continuous functions $f:\mathbb{K} \to \mathbb{R}$ whose support $\text{supp}(f) := ...
2
votes
1answer
19 views

Why are these two definitions of Lebesgue outer measure equivalent?

I got two definitions of Lebesgue outer measure one is $$\mu^*(A)=\inf\Big\{\sum^{\infty}_{k=1}l(I_k):A\subset \bigcup_k I_k\Big\}$$ and the other is $$\mu^*(A)=\inf\Big\{\sum^{\infty}_{j=1} ...
1
vote
3answers
38 views

Prove if $E_1$ and $E_2$ are measurable then $m(E_1\cup E_2)+m(E_2\cap E_2)=m(E_1)+m(E_2)$

by additivity $m(E_1\cup E_2)=m(E_1)+m(E_2)$ (because $E_1,E_2$ are measurable) but i don't know what to do with $E_1\cap E_2$. I tried to use demorgan's identity to solve this part but this is not ...
1
vote
1answer
40 views

Problem on measure theory

Let $ \mu $ be a $ \sigma $-finite measure space on $(X,s)$. Suppose $ f: X \to [0,\infty]$ be a $ s $-measurable and $ p \in [0,\infty]$. Show that $$ \int_X f^p \, d\mu = \int_{0}^{\infty} pt^{p-1} ...
1
vote
1answer
15 views

Application $\pi$-$\lambda$ lemma one-sided Markov shift

Let $(S_k^{\mathbb{N}},\Sigma_k^{\mathbb{N}},m,\tau)$ be the probability preserving transformation of the one-sided Markov shift, where $\Sigma_k^{\mathbb{N}}$ is the $\sigma$-algebra generated by the ...
2
votes
0answers
18 views

pythagorean theorem for conditional experience

Let G be a subsigma algebra and X is squareintegrable: => $ E[X²] = E[(X-E[X|G])²] + E[E[X|G]²]$ I know that this can be directly shown interpreting the conditional experience as a projection in ...
0
votes
0answers
42 views

Conditional expectation over a convex set

Let $\boldsymbol{X}$ be an $\mathbb{R}^d$-valued absolutely continuous and integrable random vector. Further, let the cdf $F$ of $\boldsymbol{X}$ be strictly increasing in each component on ...
1
vote
2answers
23 views

$\sigma$-algebra generated by the set $A:=\{n,n+1,n+2\}$ where $n\in \Bbb{N}-\{1\}$.

I am taking a course on measure theory and I have some issues regarding $\sigma$-algebra. Here is the exercise, we are taking place on $\Bbb{N}$ and I need to characterize the $\sigma$-algebra ...
2
votes
1answer
29 views

Order the domain so that function is monotonic

Let $f : \mathbb{R} \to \mathbb{R}$ be a measurable function. Is there a bijection $b: \mathbb{R} \to \mathbb{R}$ such that $f \circ b$ is monotonic?
0
votes
0answers
18 views

Difference between measure zero and volume zero?

I have the following definitions for a set to have measure zero and for a set to have volume, respectively: A set $A$ has measure zero if for any $\epsilon > 0$ there is a covering $\{S_i\}_{i \in ...
0
votes
1answer
22 views

Convergence in measure, almost everywhere and almost uniformly.

I want to find counterexamples for this: $f_n\to f$ in measure implies $f_n\to f$ almost uniformly. $f_n\to f$ in measure implies $f_n\to f$ almost everywhere. I proved that almost uniformly ...
0
votes
0answers
11 views

Decomposition of a measure

Let μ be the Lebesgue-Stieltjes measure on R corresponding to the distribution function, F where F(x) = 0 if x<0 x+1 if 0<= x<1 2x+3 if 1<= x<2 8 if ...
2
votes
1answer
19 views

Convergence in measure of product of convergent sequences

Let $(X,\Sigma,\mu)$ be a finite measurable space ($\mu(X)<\infty$). Suppose $f_n \xrightarrow{\mu} f$ and $g_n \xrightarrow{\mu} f$, prove that $f_ng_n \xrightarrow{\mu} fg$ I'll write what I ...
2
votes
2answers
37 views

Computing limits using Monotone Convergence theorem

I am trying to compute the limits of $\lim_{n \rightarrow \infty} \int\limits_0^{\infty} \dfrac{1}{(1+\dfrac{x}{n})^n \sqrt[n]x}dx $ by using Monotone convergence theorem of integrals and switching ...
0
votes
0answers
24 views

What's the difference between a random variable and a measurable function?

I've tried to wrap my head around the measure theoretical definition of a random variable for a couple of days now. In his book Probability and Stochastics, Erhan Çinlar defines a measurable function ...
0
votes
0answers
49 views

Measure theory theorem [on hold]

So far I couldn't find theorems about equality of measures, I would appreciate book recommendations and help with this theorem. Let A be a family of subsets of Ω stable under intersection. If ...
0
votes
0answers
24 views

Integration with respect to variation

Can somebody please give me a reference to what concepts are used here (I think it might be lebesgue stieltjes integration but I can't find the instance of an integration of a function with respect to ...
3
votes
1answer
28 views

second Borel–Cantelli lemma (or converse result) application

I'm having issues with understanding how Borel-Cantelli lemma applies to the following exercise: If a coin is tossed infinite times, prove that the probability of getting 2 consecutive heads (or ...
2
votes
0answers
37 views

(soft question) why do we study complex measures and complex-valued functionals in modern analysis

Recently I am struggling with "complex" things for my "real" analysis class. We are using Folland's Real analysis, 2nd for text book. It seems that Folland is trying to use complex-valued functions ...
1
vote
1answer
30 views

Integral relations when using different measures.

Let $(X,\mathcal{M})$ a measurable space and $\mu$,$\nu$ two non-negative measures s.t $\mu \geq \nu$. Does it hold that $\int_E f \, d\mu \geq \int_E f \,d\nu $ where $E \in \mathcal{M}$. I suspect ...
3
votes
1answer
34 views

If $\mu(A_n)\to 0$ then $\int_{A_n}f\to 0$

Let $(X,\Sigma,\mu)$ a measure space and $f\in L_p$, where $p\in [1,+\infty)$. Let $(A_n)$ be a sequence in $\Sigma$ such that $\mu(A_n)\to 0$. Then I want to prove that $\int_{A_n}fd\mu\to 0$. I ...
0
votes
0answers
35 views

Approximating simple function by continuous function

I am trying to solve this problem: If $\gamma$ is a simple function defined on $E \subset \mathbb R^d$, $E$ measurable, then there is $f:E \to \mathbb R$ continuous such that $$|\{x \in E: f(x) \neq ...
0
votes
1answer
31 views

How can I prove that $\int_X\left(\int_Y f_xdm_2\right)dm_1$ exists given the following conditions …?

Let $X=Y=[0,1)$ and $f(x,y)=\dfrac{1}{(1-xy)^a}$, where $a>0$, and $m_1=m_2$ the Lebesgue measure. I want to prove that $$\displaystyle\int_X\left(\int_Y f_xdm_2\right)dm_1$$ exists (the integral ...
1
vote
2answers
27 views

How can the Hausdorff measure be nonzero?

We have dim$F := \inf \left\{s > 0 : \mathcal{H}^s (F) = 0\right\}$. My question is, with dim$F$ defined as the value where the Hausdorff measure equals zero, then how can ...
2
votes
0answers
14 views

when is the maximum likelihood estimator measurable

For a random variable $X$, a class of probability measures $P_\theta$ for $\theta\in \Theta$ and their densities $f_{\theta}$ w.r.t. some common measure $\mu$, we can define the maximum likelihood ...
0
votes
1answer
16 views

Monotone functions and distribution functions

I found this quote in a textbook on measure theory I'm studying: Let $f:[a,b] \to \mathbb{R}$ be an increasing function. Since $f$ has only countably many discontinuities, we may assume without ...
0
votes
1answer
20 views

Is every $\sigma$-algebra also a semi-ring?

I wanted to know if every sigma algebra is a semi-ring? Looks like to me it is implied by the definition of a semi-ring. However I read many books and none of the books state that it is true.
1
vote
3answers
40 views

Counterexamples and convergences

I want to find counterexamples for the following "states": If $\int f_n\to \int f$ then $\int |f_n-f|\to 0$. If $\int |f_n-f|\to 0$ then $f_n\to f$ almost everywhere. Can you give me a hint of ...
1
vote
1answer
30 views

Characteristic function approximated by continuous function

I am trying to do the following problem Let $E \subset \mathbb R^d$ be measurable and let $\epsilon>0$. Show that if $A \subset E$ is measurable, then there is $f:E \to \mathbb R$ continuous such ...
1
vote
0answers
27 views

Prove that $\int (\delta x)=\delta^{-d} \int f$

Let $f$ be a real-valued integrable function on $\mathbb{R}^d$. Prove that $$\int f(\delta x) = \delta^{-d} \int f.$$ I let $f(x)=\chi_E(x)=\begin{cases} 1 & \text{if }\delta x \in E \\ 0 ...
1
vote
1answer
36 views

How to show that every set with Lebesgue outer measure zero is Lebesgue measurable?

Definition of Lebesgue measurable if for each $ε>0$, there exist a closed set $F$ and an open set $G$ with $F⊂E⊂G$ such that $m$ * $(G-F)<ε$. About this problem, $F$ can be a empty set that is ...
0
votes
3answers
24 views

Borel $\sigma$-algebra defintion question

So I am studying measure theory and I have found myself struggling to fully understand the concept of the Borel $\sigma$-algebra in depth. We know that the Borel $\sigma$-algebra is the smallest ...
0
votes
1answer
25 views

Difficulty with a differentiation of measures proof

This shows up in a proof about differentiating measures. I'm having trouble figuring it out: For any $x \in \mathbb{R}^n$, let $\mathcal{C}_r(x)$ denote the set of open cubes with diameter less than ...
0
votes
0answers
7 views

Can anyone help me prove: If $U$ is an open set in $R^n$, $f:U->R^n, f\in C^1$ , $A\subset R^n, det(f')_{|intA} \neq 0.$ Then$ A\in J \implies f(A)$

Can anyone help me prove: If $U$ is an open set in $R^n$, $f:U->R^n, f\in C^1$ , $A\subset R^n, det(f')_{|intA} \neq 0.$ Then $ A\in J \implies f(A)\in J$; $J$- set are Jordan measurable sets in ...
2
votes
2answers
51 views

If $f, g \in L^p$, is it true that $\int | f g | = \int | f | \int | g |$?

Let $f,g \in L^p(0, 1), \;\; 1 < p < \infty$. In this case, is it true that $$\underset{(0, 1)}{\int} | f(x) g(x) | dx = \underset{(0, 1)}{\int} | f(x) | dx \underset{(0, 1)}{\int} | g(x) | dx? ...
0
votes
0answers
15 views

Product of product-measurable function and measurable function product-measurable?

Given two measurable spaces $(\Omega, \mathcal{F}), (\Theta, \mathcal{F}_\Theta)$ and their product with the product-sigma-algebra $(\Omega \times \Theta, \mathcal{F} \otimes \mathcal{F}_\Theta)$ and ...
0
votes
2answers
22 views

Does a.e. convergence imply the boundness in $L^1$?

Let $f_n : I = (0, 1) \to \mathbb{R}$ be a sequence of functions. If $$f_n \to 0 \;\; a.e$$ does it imply that $$f_n \;\; \text{is bounded in} \;\; L^1(I)?$$ Why yes/not? Thank you!
2
votes
1answer
44 views

The Lebesgue-Borel measuref the difference between two open balls tends to $0$ as the radii tend to $\infty$

Let $\lambda_n$ be the Lebesgue-Borel measure on the Borel-$\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ and $x,y\in\mathbb{R}^n$. What is the easiest way to prove $$\frac ...
1
vote
1answer
31 views

What does it mean that a sequence of functions is bounded in $L^1(I)$?

Let $I = (0, 1)$ and $f_n : I \to \mathbb{R}$ a sequence of functions. What does it mean that $f_n$ is bounded in $L^1(I)$? Does it mean that $$\exists c>0 \;\; \text{such that} \;\; \|f_n\|_1 ...
0
votes
0answers
12 views

Prove for measures $\mu $and $\nu$ $\nu \perp \mu$ iff $|\nu| \perp \mu$ iff $\nu^+ \perp \mu$ and $\nu^- \perp \mu$

Where $\perp$ means mutually singular. I have a question, as $\nu$ is clearly a signed measure do we assume that $\mu$ is signed or just positive? It follows from $\nu\perp\mu$ with the set in ...
0
votes
0answers
16 views

Algebra and $\sigma$-algebra

We consider 3 intervals $A_1$, $A_2$ and $A_3$, which are defined as $$ A_1=\left(-\infty,0\right], ~A_2=\left(0,\frac{1}{2}\right], ~A_3=\left(\frac{1}{2},+\infty\right). $$ We then form the ...
1
vote
0answers
26 views

A set of the second category has a positive measure?

A subset of a topological space $X$ is called nowhere dense in $X$ if the interior of its closure is empty. A subset of a topological space $X$ is called the first category (or meagre) in $X$ if it ...
0
votes
1answer
23 views

Doubt on Caratheodory's extension theorem

This doubt is on the Caratheodory's extension in Billingsley. The main theorem says that a countably additive probability measure $P$ on a algebra can be extended to a countability additive ...
1
vote
0answers
31 views

Proof of Hunt's Interpolation

I'm new to weak $L^p$ spaces and I'm doing a book exercise. Can someone enlighten me on the proof of the Hunt's interpolation theorem, which goes as follows: Theorem Let $\langle \,M, \mu \, ...
0
votes
0answers
10 views

Regular measure on Borel sets

I am trying to do the following problem: Let $\mu$ be a measure defined on the Borel sets of $\mathbb R^n$ such that $\mu$ takes finite values on the compact sets. Let $\mathcal H$ be the class of ...
0
votes
1answer
24 views

Measurability of sequence of functions

Let $(f_n)_{n \in \Bbb N}$ be a sequence of measurable functions on a measure space $(X, M, \mu)$. Prove that the set $\{x \in X \; | \; \lim_n f_n(x) \text{ exists} \text{in } [-\infty, ...