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

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4
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1answer
53 views

Differentiation under integral sign?

I am trying to understand the following argument given in a text book: Suppose $f \in L^1(\mathbb R^n)$, consider the function $\hat{f}(\zeta)= \int_{\mathbb R^n} \exp(-2\pi i X.\zeta)f(X)dX$. ...
0
votes
0answers
22 views

Intuition - Asymptotic Maximum Likelihood Estimator

Maximum Likelihood Estimation is quite clear to me when it is performed on finite sample sizes. The intuition of an obtained Maximum Likelihood estimate for given data $x_{1},...,x_{n}$, $n \in ...
0
votes
3answers
33 views

Problem with σ-Algebra definition

At the moment I'm doing my first steps in the probability theory but now I have some problems with the σ-Algebra. Here is the definition: $A \subseteq \mathcal{P}(\Omega)$ is called σ-Algebra if: $ ...
1
vote
1answer
18 views

Let $f=\sum_1^n a_j \chi_{E_j}$ be a simple function, compute $\int |f|^p=\int |\sum_1^n a_j \chi_{E_j}|^p$.

Let $f=\sum_1^n a_j \chi_{E_j}$ be a simple function where $\mu(E_j)\lt \infty$. I want to show that $f\in L^p$, but I don't know how to compute $\int |f|^p=\int |\sum_1^n a_j \chi_{E_j}|^p$. How ...
0
votes
0answers
11 views

Point-free notation for (not direct) sums of functions over products of spaces.

I am writing a paper, and there are lots of expressions containing integrals of form: $$\int_{X \times Y} \phi(x) + \psi(y) d\alpha(x,y) $$ where $\phi$,$\psi$ are abstract functions and $\alpha$ is ...
0
votes
1answer
19 views

Showing an inequality in the proof of $L^p$ is a Banach space for $1\le p \lt \infty$.

This is part of the proof that $L^p$ is a Banach space from Folland's Real Analysis, but there is a part that I don't understand. Suppose $\{f_k\}\subset L^p$ and let $G_n=\sum_1^n |f_k|$ and ...
1
vote
1answer
25 views

Pure states on $C(X)$ are exactly evaluations

Let $X$ be a compact Hausdorff space. I want to show that pure states are of the form $ \phi (f) =f(x)$. By Reisz Represenation Theorem states on $C(X)$ are of the form $\phi (f)= \int fd\mu$ where ...
1
vote
0answers
27 views

Background for understanding Measure Theory - preparation [closed]

I've recently come across a dilemma, which I hope I can get some opinion on. I am currently finishing my undergrad degree in mechanical engineering. I have a great interest in statistics, especially ...
1
vote
3answers
34 views

Verification of a proof in Measure Theory

Let $m$ be the Lebesgue measure on $\Bbb R$ and $f:\Bbb R\to [0,\infty)$ be a Lebesgue integrable function. Show that $\exists $ a measurable set $E\subset [0,\infty)$ such that $m(E)\neq m(f^{-1}(E)$ ...
0
votes
1answer
50 views

Uniform continuity with integral being finite

Let $f$ be a real valued uniformly continuous function on $\mathbb{R}$ that is lebesgue integrable. Show that $\lim_{|x|\rightarrow \infty}f(x)=0$. Suppose that ...
3
votes
2answers
38 views

Integrating over a sequence of sets $A_n$ with $\mu(A_n)\to0$

I am going through the proof of the following. Let $(X,\mu)$ be a measure space and $f\colon X\to\overline{\mathbb R}$ be a measurable function with finite integral. If $A_1,A_2,\dots$ are ...
0
votes
1answer
19 views

Equality in Lebesgue integration inequality

Let $(X,\mathcal{M})$ be a measure space and $\mu$ be a positive measure on it. Let $g:X\to \mathbb{C}$ be a complex valued $\mu-$integrable function. We know that $$ \left| \int_X g d\mu \right| \leq ...
0
votes
1answer
28 views

the measure of the set of x where f(x)= ∞ is zero

Let $(X,\boldsymbol M, \mu)$ be a measure space where the measure $\mu$ is positive. Let $f : X \to \mathbb R^+$ be a measurable function such that $\int f \text d \mu < \infty$. Let $N = \{x \in ...
-2
votes
1answer
69 views

why is $\{x\mid f(x)=\infty\}$ measurable? [closed]

why is $\{x\mid f(x)=\infty\}$ measurable? I know that $\{x\mid f(x)=a\}$ is measurable when a is finite. But textbooks rarely mention $\{x\mid f(x)=\infty\}$. It seems to be the opposite from the ...
1
vote
1answer
70 views

A problem related to Lebesgue integration.

I have following two problems: Suppose $$\int_E f \, dx = 0 $$ where $ f: R \to R$ is a measurable function that is strictly positive. Show that $E$ must be a null set. Next Suppose that $E$ is a ...
0
votes
1answer
81 views

How can one rigorously treat integration over jump discontinuities?

Suppose I wish to compute $$\lim_{n \rightarrow \infty} \int_{0}^{\frac{\pi}{4}} \tan ^n x \,dx$$ We can imagine that the tangent function goes to $0$, with the exception of the point $\displaystyle ...
5
votes
1answer
133 views

How does the cardinality of the set of all probability measure on a set $X$ change according to the cardinality of $X$?

I was wondering concerning the following problem: Take $X$ as a parameter space endowed with its Borel $\sigma$-algebra. What is the cardinality of $\Delta (X)$, understood as the set of all ...
0
votes
0answers
17 views

existence of $\mu$ such that $\int{p}d\mu = \sum_{k=1}^{n}p^{(k)}(k/n)$

If $n\geq 1$, show that there is a measure $\mu$ on [0,1] such that for every polynomial $p$ of degree at most n, $$\int{p}d\mu = \sum_{k=1}^{n}p^{(k)}(k/n)$$ I think we should prove the linear ...
3
votes
2answers
30 views

$\mu$ is a $\sigma-$finite measure on and $\{E_n\}$ measurable sets. When $\nu(E)=\sum \mu(E\cap E_n)$, is $\nu$ is $\sigma$-finite?

My question stems from the following problem. Suppose $\mu$ is a $\sigma-$finite measure on $(X,M)$ and $\{E_n\}$ a sequence of measurable sets. Define $\nu$ on $M$ by $\nu(E)=\sum \mu(E\cap E_n)$. ...
1
vote
1answer
38 views

lebesgue and riemann integrals are the same for continuous functions on $[a,b]$

I have a proof in front of me which goes as follows, firstly assuming that the function $f \geq 0$ on $[a,b]$. We get a partition $a = x_0 < x_1 <....<x_n = b$ with $x_i - x_{i-1} = ...
0
votes
1answer
31 views

Subset $E$ of $\mathbb{R} $ is $m$-measurable iff for every $\epsilon>0$ $\exists$ a closed set $F\subset{E}$ such that $m(E\setminus F)<\epsilon$

Note: $m$ is an outer measure on the power set of $\mathbb{R}$. Attempt: Suppose the condition holds,choose closed sets $F_n\subset{E}$ such that $$m(E\setminus F_n)<1/n$$ for $n=1,2,\dots$ Let ...
-1
votes
0answers
18 views

Generic point for circle doubling map?

Let $X=\{0,1\}^{\mathbb{N}}$ withe product topology and consider the sigma algebra $\beta$ generated by the open sets. Let $T:X\rightarrow{}X$ be the circle doubling map viewed as the shift ...
1
vote
0answers
12 views

Necessary and sufficient condition for a process to be indepent of a sub-sigma algebra involving characteristic function

Consider the integrable random variable $X$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and the sub sigma-algebra $\mathcal{G}\subset \mathcal{F}$. I want to show that $$ \text{$X$ is ...
0
votes
1answer
20 views

Different ergodic probability measures are mutually singular

Can someone, please, give me a hint on how to demonstrate that different ergodic probability measures are mutually singular? Thank you!
0
votes
1answer
21 views

Multiple Lebesgue integrals: counting measure

I am completing an exercise on multiple Lebesgue integrals. The problem is as follows: Let $X=Y=\Bbb{N}$ and $\mathcal{A}=\mathcal{B}=\mathcal{P}\Bbb{N}$ with counting measures $\mu$ and $\nu$ on ...
0
votes
1answer
13 views

Characterizing isotropic measures

A Borel measure $\mu$ on $S^{n-1}$ is called isotropic if $$\int_{S^{n-1}} \langle \theta, x \rangle^2 d\mu(x)=\frac{\mu\left({S^{n-1}}\right)}{n}$$ for all $\theta\in S^{n-1}$. This means that in ...
0
votes
2answers
21 views

finite additive measures question

Let Ω be a countable set and F the collection of all its subsets. Put µ(A) = 0 if A is finite and µ(A) = ∞ if A is infinite. Show that the set function µ is finitely additive but not σ-additive. Does ...
0
votes
2answers
31 views

Non-measurable subset of a null set.

I am reading measure theory,and I am searching an example in which a measurable null set have a non-measurable subset because this is the reason that,s why we are studying about complete ...
1
vote
1answer
25 views

constructing sigma-field generated by certain set

i'm studying mathematical statistics. and i got a problem that finding a $\sigma$-field generated by $D$ = {A,B} the answer i heard(from professor..) is $B$ = {$\varnothing$ , A, B, A$^c$,B$^c$, ...
4
votes
1answer
64 views

Set measurable with respect to one product measure but not with respect to another

For $p \in (0,1)$, let $\mu_p$ be the measure on $\{0,1\}$ given by $\mu_p(\{1\}) = 1 - \mu_p(\{0\}) = p$. We can extend $\mu_p$ to a product measure on the countably infinite product ...
5
votes
2answers
107 views

Uncountable Borel Sets

Let $A$ be an uncountable Borel subset of $\mathbb{R}^n$, and consider the Lebesgue measure on $\mathbb{R}^n$. Assume the axiom of choice (if you need it). Does there exist a Lebesgue measurable set ...
0
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0answers
12 views

Proof verification: $\frac{d(\nu_1\times \nu_2)}{d(\nu_1 \times \nu_2)}(x_1,x_2)=\frac{d\nu_1}{d\mu_1)}(x_1)\frac{d\nu_2}{d\mu_2}(x_2).$

This is exercise 3.12 from Folland's Real Analysis. It took me a long times to come up with a solution to this problem, and I'd appreciate it if anyone could verify if my answer is correct. For ...
0
votes
1answer
33 views

Construct sigma-algebra ( sigma-field)

i'm studying mathematical statistics, with hogg, while constructing sigma-field generated by $D$, where $D$={C,D,E,F} i don't know what is sigma-filed generated by $D$. let's say universal set $X$ ...
0
votes
0answers
31 views

A measure for P(R)

We know that -unfortunately- Lebesgue measure cannot measure all the subsets for $\mathbb{R}$ for example (Vitali Set) My question is : can we build a measure for all the subsets of $\mathbb{R}$, ...
1
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0answers
17 views

The definition of completion of one measure with respect to a family

The following is taken from the book of Revuz and Yor more or less verbatim. If $(E,\mathcal E)$ is a measure space carrying probability measure $\mu$, the completion $\mathcal E^\mu$ is the ...
2
votes
1answer
54 views

Showing the outer lebesgue measure of a set is $0$

Let $A$ be the set which consists of the real numbers $x$ that can be written in the form $$x = \sum_{i=1}^\infty a_i/10^i$$ where each $a_i \in \{1,\dots,9\}$ show that $\lambda^*(A) = 0$. I have ...
1
vote
1answer
19 views

Is this outer measure really regular?

Definition of regular according to Halmos: Let $\textbf{H}$ be a hereditary $\sigma$-ring (i.e. a $\sigma$-ring such that if $F \in \textbf{H}$ and $E \subset F$ then $E \in \textbf{H}$) and $\mu^*$ ...
0
votes
1answer
37 views

Two questions on Lebesgue integration and application of reverse triangle inequality

I'm learning about measure theory (specifically Lebesgue integration) and need help with the following problem: Let $f_n, f \in L^1$ and $\int_{\mathbb{R}}\left|f_n-f\right| \rightarrow0$. Prove ...
0
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0answers
18 views

Measurable sets - Outer measure on a hereditary $\sigma$ ring $H$

Let $\mu^{*}$ be an outer measure on a hereditary $\sigma$-Ring $H$. A set $E\in H$ is $\mu^{*}$- measurable if, $\forall A\in H$ $\mu^{*}(A)=\mu^{*}(A \cap E)+\mu^{*}(A\cap E')$. But $E'$ does not ...
1
vote
1answer
24 views

$\int_{\bigcup_{n=1}^{\infty}E_n}f=\sum_{n=1}^{\infty}\int_{E_n}f$ given $f$ positive and measurable

I'm learning about measure theory (specifically Lebesgue intregation) and need help with the following problem: Let $f:\mathbb{R}\rightarrow[0,+\infty)$ be measurable and let $\{E_n\}$ be a ...
1
vote
3answers
30 views

Countable subset of range of $\mu$-measurable function

I am reading through the proof of the following. If $f\colon X\to\overline{\mathbb R}$ is $\mu$-measurable for some $\sigma$-finite measure $\mu$, then the set $\{r\in\mathbb R\mid \mu(\{x\mid ...
3
votes
1answer
31 views

Martingale convergence for UI martingales

I started reading this paper (Lamb, Charles W.. “Shorter Notes: A Short Proof of the Martingale Convergence Theorem”. Proceedings of the American Mathematical Society 38.1 (1973): 215–217) today. In ...
4
votes
0answers
129 views

Surface area from indicator function

I know that the volume and the surface area of a sphere of radius $R$ are related by a derivative: $$V(R)=\frac{4}{3}\pi R^3$$ $$A(R)=4\pi R^2=\frac{\partial V(R)}{\partial R}$$ I am asking if an ...
0
votes
0answers
42 views

construct a set $H$ such that $S\cup H$ is open and $\lambda(H) < \epsilon$

Let S be a lebesgue measurable subset of R and let lambda denote the lebesgue measure. Show that there exists sets $H$ and $K$ such that 1) $S \cup H$ is open and $\lambda(H) < \epsilon$. 2) $S ...
0
votes
2answers
89 views

if $f_n \to f$ and $f_n$ are measurable, then show $f$ is measurable

the solution writes, $\{ x: f(x) > c \} = \cup_n \{x: f_n(x) > c\}$, then concludes. But surely this isn't true, as if $f(x) > c$ this means that there is a number $N$ such that for all ...
0
votes
1answer
29 views

Prove that if h is an integrable function, then $ \lim_{c \to 0}$ of the integral over the real line of $|h(x)-h(x+c)| = 0$

Prove that if h is an integrable function, then the $ \lim_{c \to 0}$ of the integral over the real line of $|h(x)-h(x+c)| = 0$ For this problem would I just say that $h(x)= h(x+c)$ almost everywhere ...
0
votes
1answer
34 views

Show the sigma algebra containing $\{2n,2n+2,2n+4,..\}$ is uncountable

I am asked the following: 1) Show that for any family of subsets $A$ of $\Omega$ there is a smallest $\sigma - $algebra containing $A$. 2) Let $\mathcal{F}$ be a sigma algebra in the space $\Omega$ ...
1
vote
1answer
23 views

short problem on the use of radon nikodym derivative

suppose $\mu$ and $\nu$ and measures such that $\nu(\Omega) = 2$. Let $f$ be the Radon-Nikodym derivative of $\mu$ with respect to $\mu + \nu$. Find $\nu(\{x: f(x) < 1\})$ Let $F = \{x: f(x) \geq ...
1
vote
2answers
40 views

Show that a function is not integrable

show that $f(x,y) = \dfrac{2xy}{1+x^4+y^4}$ is not $\lambda_2$ integrable. I am given the solution, and it states: $$\int f^+ \ d \lambda_2 \geq \int_{(0,\infty)^2} f^+ d \lambda_2 = \int_0^\infty ...
0
votes
1answer
13 views

Showing a $2$-dim function is measurable

Give the definition of the sigma-algebra $B_2$ of Borel measurable sets in $\mathbb R^2$. If $\mathbb G$ is the collection of all open subsets off $\mathbb R^2$, then the sigma-algebra generated by ...