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

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3
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3answers
107 views

When $f(x) = g(y)$ for almost every $(x,y)$, must $f$ and $g$ be constant almost everywhere?

Consider two measure spaces $(X,\mathcal{A},\mu)$ and $(Y,\mathcal{B},\nu)$, where $\mu\times\nu(X\times Y)>0$. Given two measurable functions $f:X\to \mathbb{R}$ and $g:Y\to\mathbb{R}$ such that ...
0
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0answers
20 views

Jordan Measure of Ascending Union

By definition, Jordan outer measure of a subset $E$ in $\mathbb{R}^n$ is the approximation to area of $E$ by finitely many open cubes(rectangles) which cover $E$. Similarly, the Jordan inner measure ...
0
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0answers
28 views

Measuring Unsigned Simple Functions

I was hoping that someone would be able to help me solve this problem regarding simple functions and their measure. This problem is coming straight from Introduction to Measure Theory by Terrence Tao. ...
0
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1answer
24 views

Borel-Cantelli Theorem

The following is a problem from Stei-Shakarchi's Real Analysis: Suppose $(E_n)$ be a countable family of measurable sets such that $\sum_n m(E_n)<\infty$. Define $E=\{ x\in\mathbb{R}^d\colon x\in ...
0
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0answers
13 views

Why all closed intervals of $R$ is a semi-algebra?

How the class of all closed intervals can be a semi-algebra?
3
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1answer
46 views

Inducing a surface area measure on $S^2$ from the Haar measure on $SO(3)$

I'm reading the book "Random Matrices: High Dimensional Phenomena" by G. Blower. There is an example that I've been struggled for a long time. For those who have access to the book, it's the Example ...
1
vote
1answer
19 views

proving continuity of decreasing measurable sets, without using same results for increasing measurable sets

There is a well known result in measure theory that says that: Suppose that $(\Omega,A, \mu)$ is a measure space. If $\{E_n\}_{n=1}^\infty\subseteq A$, with $E_1 \supset E_2...$, and $\mu(E_1) ...
2
votes
1answer
39 views

Fourier coefficients of a (finite, regular, positive) measure are absolutely summable => the measure has a density

Let $\mu$ be a finite, regular, positive measure on $[0,1)$ such that $\sum_{n\in\mathbb{Z}} |\hat{\mu}(n)| < \infty$. How can I prove that there exists $f(x)$ such that $\mu(dx) = f(x)dx$? ...
0
votes
1answer
16 views

Factor theorem for $\bar {\mathcal M}(\mathcal E)^+$ (set of $\mathcal E$-$\mathcal B(\mathbb R)$-measurable functions with values in $[0,\infty]$).

Factor theorem for $\bar {\mathcal M}(\mathcal E)^+$ (set of $\mathcal E$-$\mathcal B(\mathbb R)$-measurable functions with values in $[0,\infty]$). Let $X$ be a non-empty set, let $(Y,\mathcal ...
0
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0answers
13 views

Extension of Premeasures

Here, a premeasure is a countably additive set function whereas a measure is one acting on a sigma-algebra. Not every positive premeasure admits an extension to a positive measure as the following ...
0
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1answer
34 views

Does the class of all finite unions of closed-open intervals on $\mathbb{R}$ form a ring sets?

Does the class of all finite unions of closed-open intervals on $\mathbb{R}$ form a ring on sets? By a closed-open interval , I mean an interval of the form $[x,y)$ A ring of sets is a non-empty ...
1
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1answer
29 views

Why $f (x):= \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)}$ only belongs to $L^2(0, \infty)$

This is a result given in Royden and Fitzpatrick (p. 143). Show that $$ \int_0^\infty \left[ \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)} \right]^p < \infty $$ if and only if $p=2$. That ...
1
vote
2answers
20 views

Measures: Sequential Continuity

Disclaimer: This thread is meant as record and written in Q&A style. Let $\Omega$ be a finite measure space $\mu(\Omega)<\infty$. It is well known that a measure is continuous from above as ...
5
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0answers
52 views
+50

Question about B. Host paper 'Nombres, normaux entropie, translations'

I was reading this paper and I got stuck in a detail left for the reader that I couldn't figure out: Let $X = \mathbb{R}/\mathbb{Z}$, $p > 1$ a integer, $D_n = \{kp^{-n}\colon 0 \leq k < p^n ...
0
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1answer
20 views

Vitali Set: Inner Measure vs. Outer Measure

Context Nonlinearity in general of the Lebesgue integral for nonmeasurable functions reduces in some sense to inner and outer measure of nonmeasurable sets: ...
0
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0answers
19 views

Counting the exact number of sets in the Borel Field generated by a collection of “unrelated” sets

Prove: The B.F. generated by n given sets "without relations among them" has $2^{(2^n)}$ members. To be perfectly clear, "without relations among them" means that no set in the generating ...
0
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0answers
28 views

Uniform Wiener-Wintner Theorem - proof

I am looking for proof of uniform version of Wiener-Wintner theorem: Let $(X, \mathcal{A}, \mu, T)$ be an ergodic measure preserving system. For $f \in L^1(\mu)$ which is orthogonal to the ...
1
vote
1answer
21 views

Does this proof for the MCT hold for the extended real valued functions.

Here is a proof for the MCT, but it says that it is for the real numbers, not the extended real numbers. If we allow the function f to take the value infinity does the proof still hold? I can not see ...
1
vote
1answer
17 views

Order between probability measures: sets full below

Consider a product space $X = \{0,1\}^\mathbb{Z}$ and the space of probability measures on $X$, $\mathcal{M}(X)$. We say that for any two $a, b \in X$, $$a \prec b \iff a_x \leq b_x \, \, \, \, \, ...
1
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1answer
27 views

Prove $\int s d \mu = \sum^n_{j=1} a_j \mu(A_j)$ for $s=\sum^n_{j=1} a_j 1_{A_j}$ not a standard representation of $s$.

Let $(X, \mathcal E, \mu)$ be a measure space. Let $s \in \mathcal S\mathcal M(\mathcal E)^+$ be a simple function written as $s= \sum^n_{j=1} a_j 1_{A_j}$ , $a_j \ge 0, A_j \in \mathcal E$. Prove ...
0
votes
0answers
32 views

Condition for a function $f: \mathbb R \rightarrow \mathbb R$ being right or left-continuous at $a \in \mathbb R$.

I know that $f: X \rightarrow \mathbb C$ is continuous if and only if for every convergent sequence $(x_n)$ in $X$ the identity holds $\lim_{n \rightarrow \infty} f(x_n) = f(\lim_{n \rightarrow ...
0
votes
1answer
19 views

THE sigma-ring or A sigma-ring?

I have two questions about sigma-rings and measure spaces. Let $(\Omega, \mathscr{F}, \mu)$ be any measure space. Is $\mathscr{F}$ THE sigma-ring of this space or A sigma-ring of this space? If ...
2
votes
1answer
22 views

Measure theory: proof of the “Standardproof” given theorem.

Measure theory: proof of the "Standardproof" given theorem. Let $(X, \mathcal E)$ be a measurable space. Let $W \subseteq \mathcal M(\mathcal E)$ (set of measurable $\mathcal E$-$\mathcal ...
0
votes
1answer
17 views

How to do you compute the probability a record occurring in a sequence of independent experiments?

Consider a sequence of independent experiments, each of which produces a random integer in N with the probability mass function ${p_k}$. The pmf is the same for all the experiments and also strictly ...
1
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0answers
18 views

Can simple functions take the value infinity?

I don't think my book is clear about this. It is "a course in real analysis", by weiss. Now I am in the chapter about the general lebesge integral, and we are going to develop the non-negative ...
1
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0answers
26 views

The non-existence of one distribution

The problem is to prove that does not exists a distribution $u$ on $\mathbb{R}$ such that $$ \langle u, \varphi \rangle = \int e^{1/x^2} \varphi(x) \, dx, \hspace{0.9cm} \varphi \in ...
0
votes
0answers
30 views

About counting measure on Borel sets

Let $\mathcal{C}$ be all finite unions of half open intervals, $\mathcal{A}=\sigma(\mathcal{C})$, i.e., the Borel $\sigma$-algebra. Suppose that $\mu$ is the counting measure, and $\nu=2\mu$. Can ...
0
votes
1answer
36 views

Problem with topological space in probability theory. [closed]

Let $(X, \tau)$ be a topological space. a) Show that arbitrary intersections of closed sets are closed. b) Prove that a set $F \subseteq X$ is closed if and only if for all sequences $\{x_{n}\} ...
2
votes
1answer
55 views

Is this description of “sigma-algebra generated by collection of subsets” right?

Disclaimer: sorry for my poor english and edition. Claim: If $M\subseteq \mathcal{P}(X)$, then $\Sigma(M)=M_3$, where: $\Sigma(M)$ is the sigma-algebra generated by $M$ $M_1=\{A\subseteq X:(A\in ...
1
vote
3answers
42 views

How can I prove that $\{ \ (x,y)\in \mathbb R^2 : x >0, 0\le y \le 1/x \ \} \in \mathcal B(\mathbb R^2)$ is a Borel-set in $\mathbb R^2$?

How can I prove that $\{ \ (x,y)\in \mathbb R^2 : x >0, 0\le y \le 1/x \ \} \in \mathcal B(\mathbb R^2)$ is a Borel-set in $\mathbb R^2$ ? I have tried to construct this set from countably union ...
0
votes
0answers
17 views

Help me in this excercise with Hermetian scalar product. [on hold]

On the vector space $C[-1,1]$ is the Hermetian scalar product $(f,g):=\int_{-1}^{1}f(x)\overline{g}(x)dx$ defined. a)Determine the function in W = span{1,x}, closest to $f(x)=x^3$ is. b)Determine ...
1
vote
1answer
22 views

Measure defined in an atypical way

I was reading a paper when I found this ($\partial \Omega$ refers to the boundary of $\Omega$ and $\nabla$ to the gradient operator,$\nabla f = (\partial_{i}f)_{i} $ ). Let $\Omega \subset ...
0
votes
1answer
40 views

Borel measurable functions

Suppose that f:R→R is a Borel measurable function and let h:R^2→R be defined by h(x,y)=f(x)+f(y). Prove that h is Borel measurable
1
vote
1answer
31 views

Operator induced by continuous function and measures

If $X$ is a compact metric space, and $T:X \rightarrow X$ is continuous map, what would be meant by $T_\ast$ is the operator on measures induced by $T$? Allow $\mu$ to be some Borel regular normed ...
1
vote
1answer
15 views

How does one prove that elements of the Borel set are regular?

How does one prove that elements of the Borel set are regular? A Borel set of course being any element of the Borel sigma algebra (say A), and regular meaning that for a given real number e, there is ...
0
votes
1answer
24 views

$\sigma$-field generate by one point sets

I have the following problem. Let $\Omega$ be a non-empty set and let $\mathcal{C}$ be all one point subsets. Show that \begin{align} \sigma(\mathcal{C})=\{A \subset \Omega : A \text{ is countable ...
0
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0answers
36 views

Prove that $A$ is Lebesgue measurable implies that $x+A = \{ x+y : y \in A \}$ is measurable

This question comes from Exercise 4.5 of Real Analysis for Graduate Students by Bass. After some deduction I reduced the question to the following form: Show that if $A$ is a Lebesgue measurable set ...
0
votes
1answer
33 views

Ask for a good reference for the calculus involving singular continuous measure

I am not an expert on measure theory. I am sorry if this question is too simple for some experts here. Suppose the measure $\mu$ is singular continuous on $\mathbb{R}$, such as the cantor measure. ...
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votes
2answers
34 views

Show that $\mathcal{A}$ is a $\sigma$-field [closed]

Suppose that $\mathcal{A}$ is a field and suppose also that $\mathcal{A}$ has the property that it is closed under countable disjoint unions. Show that $\mathcal{A}$ is a $\sigma$-field.
2
votes
1answer
25 views

Elementary question about the Lebesgue-measure $\lambda$

Formally speaking, the Lebesgue-measure $\lambda$ on $\mathbb{R}$ is the restriction of the outer Lebesgue-measure $$ \lambda^*(A)=\inf\left\{\sum_{n=1}^{\infty}p(C_n): ...
0
votes
1answer
28 views

Measure of region

Let $\Omega:=[0,1]^2$, $f(x):=-x+1$ and $g(x):=(x-1)^2$. I am supposed to compute the $L^2$ measure of the area of the region given by $$M:=\{(x,y)\in\Omega\;|\;g(x)\leq y\leq f(x)\}.$$ Can I just ...
6
votes
2answers
68 views

Prove Borel Measurable Set A with the following property has measure 0.

This question is exercise 4.10 of Richard F. Bass's Real Analysis for Graduate Students, 2nd edition. Let $\epsilon \in (0,1)$, let m be Lebesgue measure, and suppose A is Borel Measurable subset of ...
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0answers
23 views

Sum of two measurable sets

I have heard that sum of two Lebesgue measurable sets in $\mathbb{R}$ may not be Lebesgue measurable. Can anyone give me an example with explanation?
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3answers
52 views

A and B are independent under$\mathbb{P}$ but not under $\mathbb{Q}$

As the title, how to construct such an example that 2 events from the same measurable space ($\Omega$,$\mathscr{A}$) are independent under probability measure $\mathbb{P}$ but not independent under ...
3
votes
1answer
27 views

computing the Haar measure for O(n) and U(n) groups

My question is about how to compute the Haar measure for O(n) and U(n) groups. For example, for the conventional parametrization of SO(3) with 3 angels, the Haar measure is $ dO= ...
2
votes
4answers
426 views

What does it really mean when we say that the probability of something is zero? [duplicate]

Conventionally, people will say a probability of zero is equivalent as saying that the event is impossible. But when we look at the probability from a mathematics perspective, probability is defined ...
0
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1answer
31 views

complex measurable functions

I am trying to prove something about complex measurable functions. I have an idea for one direction and hope someone can give me a hint, I have gotten somee work done in this direction but need help ...
4
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0answers
61 views

A Question From Measure Theory

How to show that a basis for the vector space $\mathbb{R}$ over the field $\mathbb{Q}$ is not Lebesgue measurable? Can anyone help me?
4
votes
1answer
43 views

Can measure induce a topology on a Set?

When I was taught metric spaces in Topology, I came across the idea that metric defined on a set can induce a topology by creating a basis (open balls). If we have a measure defined on a set, can it ...
0
votes
2answers
27 views

Describe the sigma algebra generated by singleton subsets

Let's denote the set of all singleton subsets of $X$(i.e. of all subsets consisting of one element) by $A$. Describe $\sigma(A)$ in the following two cases: i) $X$ is countable ii) $X$ is ...