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

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4
votes
1answer
30 views

Question on Lebesgue integration and partition?

$A_1, A_2, \ldots, A_n$ are measurable sets w.r.t $(X,\mathcal A,\mu)$ with $\mu(X)<\infty$. If each $x\in X$ belongs to at least $q$ sets in $A_1, A_2, \ldots, A_n$ then there exits a set $A_i$ ...
2
votes
2answers
50 views

If $f$ is Lebesgue integrable on $ R$ then $\lim_{h\to 0} \int_R |f(x+h)-f(x)|dx=0? $

If $f$ is Lebesgue integrable on $R$ then $\lim_{h\to 0} \int_R |f(x+h)-f(x)|dx=0$ My attempt: I am trying to use the definition of Lebesgue integration but I am stuck. Can anyone give me some ...
5
votes
1answer
41 views

For Lebesgue integrable $f$, show $\lim_{t\to +\infty}t\mu(\{x : |f(x)| \ge t \})=0$

Function $f$ is integrable on set $A$ w.r.t measure $\mu$ and $A_t=\{x\in A: |f(x)|\geq t\}$. Show that $\lim_{t\to \infty} t\mu(A_t)=0.$ I think that question is false, should $\lim_{t\to ...
6
votes
3answers
91 views

Question on Lebesgue integration?

Define $$\int_{E^*}f\ \mathsf d\mu = \inf_{\{A_i\}} \sum_i\left[\sup\limits_{x\in A_i}f(x)\right]\mu(A_i),$$ where $\{A_i\}$ is a partition of $E$. I want to prove that if $$\mu(\{x: f(x)>0\}) ...
0
votes
0answers
27 views

Does this function have a name?

The following function came up when I was working on a problem and I want to know if there is anything known about it. Do you know if this has a name? $p_{E}(x) := \lambda(\{z \in E: d(z,y) = x$ for ...
3
votes
2answers
31 views

Show that $\mathbb{E}[X_t]=X_0e^{-ct}$ if $X_t=X_0e^{-ct}+\sigma e^{-ct}\int_0^te^{cs}dW_s$, $X_0\in\mathbb{R}$

so I know the result is trivially correct, but I am being asked to prove it. I tried using a theorem, but it seems rather contradictory. Thanks in advance! Question: Show that ...
2
votes
1answer
35 views

What is the Haar measures on $SL(2, R$ And $SL(2,R) / SL(2, Z)$?

How does one parametrize those spaces in order to do integration over them? What's a good reference for doing integral a with Haar measures over matrix groups?
5
votes
2answers
76 views

Boundary of closure of open set in $\mathbb R^2$ has measure zero

Consider problem 4 on day 2 of this exam. Suppose that $\mathcal O\subset \mathbb R^2$ is an open set with finite Lebesgue measure. Prove that the boundary of the closure of $\mathcal O$ has ...
2
votes
0answers
91 views

Is “almost all function” a well defined concept?

I am working on a problem which has well defined properties for the vast majority of all PDFs. I would like to make a quantitative statement along the lines of "for almost all distributions, P holds". ...
2
votes
1answer
41 views

roots of polynomial with measurable coefficients

If I have polynomial: $$a_n(\omega) x^n+\cdots+a_0(\omega)$$ $a_i:\Omega\to\mathbb R$ are measurable functions assume that for every $\omega$ there are $n$ real roots to the polynomial (including ...
0
votes
1answer
27 views

How to show that $P(X_{n} \geq n$ $i.o.) = 0$, given $E(X_{i}) = 0$ and $E((X_{i})^{2})=1$ for $i=1,2,3…$

I'm struggling with the following problem (Exercise 4.5.16 in Rosenthal's probability book): Let $X_{1}, X_{2},...$ be defined jointly on some probability space, with $E(X_{i}) = 0$ and ...
1
vote
1answer
52 views

Why is $\mathfrak{L}^1$ not a vector space of functions that take infinity as a value

In measure and integration theory we sometimes work with measurable functions $$ f: (X, \mathfrak{A}) \to (\overline{\mathbb{R}}, \overline{\mathfrak{B}}),$$ where $(X,\mathfrak{A})$ denotes an ...
2
votes
1answer
65 views

Proof the additivity of a measure.

Theorem : If $\{A_{1},A_{2},A_{3},....\}$ is a countable disjoint collection of measurable sets, then $$\mu \left(\bigcup_{i=1}^{\infty} A_{i}\right) = \sum_{i=1}^{\infty} \mu (A_{i}).$$ ...
2
votes
1answer
21 views

Dirac Delta Function as a Measure

I was always told in my college physics classes to not worry too much about the dirac delta function because it can be made rigorous using distributions or measure theory. I've just started learning ...
1
vote
0answers
50 views

Prove the integral $\int_{1}^{\infty} \ln(1+ \frac{(\sin x)^n}{x^c})dx$ exist as a finite Lebesgue integral

For what nonnegative integer $n$ and positive real $c$ does the integral $\int_{1}^{\infty} \ln(1+ \frac{(\sin x)^n}{x^c})dx$ exist as a finite Lebesgue integral and when does it converge as an ...
2
votes
2answers
79 views

Prove that $\limsup_{n \rightarrow \infty} (f_n(x))^{1/n} \leq1 $

Suppose $f_n$ are nonnegative measurable functions on a measure space satisfying $$\int f_n\ \mathsf d\mu =1.$$ Prove that $$\limsup_{n \rightarrow \infty} (f_n(x))^{1/n} \leq1 $$ almost ...
0
votes
1answer
41 views

Is the convolution pointwise bounded?

A problem from an old exam: Prove or disprove: if $p,q \in [1,\infty)$ such that $p^{-1}+q^{-1}=1$ and $f\in L^p, g\in L^q$, then the convolution $f*g$ is pointwise bounded. First of all: what ...
3
votes
2answers
56 views

Is the positive or negative variation of a signed measure finite?

I'm studying measure theory and read about signed measure. A signed measure is a function $\nu:\mathcal{A}\to \mathbb{R}\cup\{\pm\infty\}$, where $\mathcal A$ is a certain $\sigma-$algebra, such that ...
7
votes
2answers
63 views

Show that $\sum^\infty_{n=1} \mu(\{x : |f_n(x) - f(x)| > \epsilon\}) < \infty$ implies $f_n \to f$ a.e.

Show that $\sum^\infty_{n=1} \mu(\{x : |f_n(x) - f(x)| > \epsilon\}) < \infty$ implies $f_n \to f$ a.e, where $f_n$ and $f$ are measurable functions. My attempt: The Borel Cantelli lemma gives ...
2
votes
1answer
27 views

Radon measure times a function is still a Radon measure?

Given $\Omega\subset \mathbb R^N$ is open and let function $\varphi$: $\Omega\to [1,+\infty]$, $\varphi\in L^1_{loc}(\Omega)$ be given. Suppose $\mu$ is a finite Radon measure on $\Omega$ and we ...
1
vote
1answer
38 views

What is the interpretation of $\mu(dx)$ in Lebesgue integral?

For estimating the integral $\int_a^b f(x)dx$ we calculate the area of rectangles of height $f(x)$ and base $dx$ (Riemann sums). Therefore, we go from $a$ to $b$ with steps of $dx$. what 's the ...
1
vote
0answers
26 views

Measurability of Dini Derivatives

Let $f:(0,1)\to\mathbb R$ be measurable. Then, the (right upper) Dini derivative $$ D^+ f(x) = \limsup_{h\to 0^+} \frac{f(x+h) - f(x)}{h} $$ is also measurable (a well known result of Banach). Can ...
1
vote
1answer
33 views

How to write down the probability space of this stochastic process

Consider infinitely repeated coin-toss. Then the probability space can be written as $\Omega=\{H,T\}^\infty$ with its product $\sigma$-algebra. Now let's assume that after each round, there is ...
7
votes
2answers
131 views

Prove that jump functions are measurable

This question comes from the exercises of Stein and Shakarchi's Real analysis Ex. 5.14. Define $$ j_n(x)= \begin{cases} 0& \text{if } x< x_n\\ \theta_n & ...
0
votes
0answers
21 views

Dirac Measure is Purely Atomic

In my book, "Probability and Stochastics" by Cinlar, it's stated that for some measurable space $(E,\scr E)$, and fixed $x\in E$, the Dirac measure $\delta_x(A)=\left\{ \begin{array}{lcc} ...
2
votes
3answers
66 views

Lebesgue outer measure of disjoint sets in $\mathbb{R}^n$

If $d(A, B) > 0$, then it's true that $m^*(A\cup B) = m^*(A) + m^*(B)$. If there are disjoint open sets $U, V$ such that $A \subset U$ and $B \subset V$, doesn't it still hold that $m^*(A\cup B) = ...
3
votes
2answers
101 views

Exercise of measure theory

I need help with this exercise. Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $(A_n ;n\geq 1)\subseteq \mathcal{F}$, such that $\mu (\bigcup_{n=1}^{\infty}A_n)<\infty$ and ...
0
votes
1answer
30 views

Skorokhod space with uniform norm is Banach

Let $D := D([0,t])$ be the Skorokhod space of right-continuous functions with left limits taking values in $\mathbb{R}^d$. Equip $D$ with the supremum norm $||f||_\infty = \sup_{s \in [0,t]}|f(s)|$. ...
2
votes
1answer
52 views

Why is the support of Dirac distribution $\{0\}$?

Distributions are of two types: those that are obtained from locally integrable functions, and those that aren't. For the first type, the support of distribution is simply the support of the function. ...
0
votes
1answer
67 views

Prove that this function is not Riemann-integrable

Let $A\subset[0,1]$ be the union of open intervals $(a_i,b_i)$ such that each rational number in $(0,1)$ is contained in some $(a_i,b_i)$ and $\sum_i(b_i-a_i)<1$. It can be shown that $\partial ...
1
vote
0answers
59 views

Strong law of large numbers

Suppose $X_i\in\mathcal{L}^2$ with expectation $0$ such that $\sum_{i=1}^\infty \mathbb{E}[X_i^2]/i^2<\infty$ and suppose they are pairwise non correlated. Does then the SLLN still hold?
0
votes
2answers
81 views

$f(x) > 0$ , prove that $\int{f} > 0$

Let $f$ be a function such that $ \forall{x}, f(x) > 0 $ and is integrable in $[a,b]$ prove that: $ \int_a^b{f(x)} > 0$ or show a counter example
1
vote
0answers
20 views

Properties of the Kernel from the measurable space $(X,\mathscr{A})$ to $(Y,\mathscr{B})$

Hi everyone this is an exercise from Cohn's book. I'd appreciate if someone can check part (d) and (e) where I have more problems because this concept is completely new for me. Let $(X,\mathscr{A})$ ...
0
votes
1answer
129 views

New definition of Lebesgue integral

Let $(X, \mathcal M, \mu)$ be a measure space. Let $g: X \rightarrow [0, \infty]$ be a non-negative extended real-valued function. We call $g$ an elementary function if $g$ is measurable and $g(X)$ is ...
3
votes
1answer
43 views

Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y)dy = 0$ for all $x$. Then $f = 0$ a.e.

This is a corollary to a proof in Bass, but I don't understand why it follows from the proof he gives. I follow everything up until the last statement. Why is it that proving that the integral is $0$ ...
0
votes
0answers
15 views

Doubt regarding limitting value of partial derivative of $C^{1}$ function

The aim is to prove the following result: Let $v : \mathbb R \to \mathbb R$ be such that: $v \in C^{1}(\mathbb R)$ & $|\frac{\partial v}{\partial x}| \in L^{1}(\mathbb R)$ . Then to prove that: ...
0
votes
1answer
37 views

If a set of functions contains a sequence that is Cauchy, but not convergent, what does that imply about the set or the functions?

This is not homework. I am studying a set $S$ of of distributions that all have a fixed mean and also satisfy some ancillary criteria. Lets say I have a sequence of probability density functions ...
0
votes
1answer
11 views

Interpretation of proof of the Lebesgue-Stieltjes measure as a $\sup$ over compact subsets

The following is part of a theorem and proof in Folland's Real Analysis: Modern Techniques and Their Applications: Let $\mu$ be a complete Lebesgue-Stieltjes measure on $\mathbb{R}$ associated to the ...
5
votes
1answer
80 views

Under what conditions does a specified conditional distribution exist

It is common to see conditional distributions specified in stats like: $$(X \mid \mu = t) \sim \mathcal{N} (t, 1)$$ (Of course, we can also use some other distribution here) How do you prove that ...
1
vote
1answer
21 views

Calculate the measure of a measurable set under nonlinear mapping.

It is known that: If $\cal{A} \subset \mathbb{R}^n$ is Lebesgue measurable, and $L: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a linear mapping, then $L(\cal{A})$ is Lebesgue measurable and ...
-1
votes
1answer
27 views

Algebraically solve a component in a partial sum

I would like to solve this equation for y: $$T = -a + \sum_{1}^{n} \frac{\left(\frac{x}{n} - \frac{y}{n} \right)}{ (1+b)^{n} }$$ The partial sum (Σ) is from 1 to n. I use the ^ symbol for an ...
2
votes
1answer
40 views

Lebesgue measure of a set of real numbers well-approximated by rationals

Consider the set $A$ of real numbers $r$ such that there exists a constant $C$ and sequence $\frac{p_n}{q_n}$ of rational numbers (where $p_n$ and $q_n$ are integers) with $q_n \rightarrow \infty$ and ...
0
votes
0answers
25 views

Integral of sup of directed family of elementary functions

Let $(X, \mathcal M, \mu)$ be a measure space. Let $g: X \rightarrow [0, \infty]$ be a non-negative extended real-valued function. We call $g$ an elementary function if $g$ is measurable and $g(X)$ is ...
3
votes
1answer
35 views

Haar measure - a problem from Folland

I was presented with this question from Folland's real analysis second edition involving Haar measures. It is problem 3 of chapter 11 page 347, which reads as follows: Let G be a locally compact ...
0
votes
1answer
36 views

Proof that outer measure of interval equals length, why use Heine-Borel?

Define by $$ m^*(A) := \inf\left\{ \sum_i |I| : A \subseteq \bigcup_i I_i \right\} $$ the outer measure of some set $A \subseteq \mathbb R$. Then we have $m^*(I) = |I|$ for each interval (open, ...
2
votes
1answer
67 views

How to find the density of $Y=g(X)$ in this case?

I have a vector $X=(1,X_2,X_3)$, where $(X_2,X_3)$ is a random vector in $\mathbb{R}^2$. Now consider $Y=g(X)=X/\|X\|$. What is a density function of $Y$ with respect to the uniform spherical ...
1
vote
0answers
23 views

Show that a collection of finite unions of sets of the form $(a,b]\cap \mathbb{Q}$ is an algebra

The following is a question from Folland's Real Analysis: Modern Techniques and their Applications. (Question 23 page 32) Let $\mathcal{A}$ be the collection of finite unions of sets of the form ...
1
vote
1answer
63 views

Exercise 43 chapter 2 in Real Analysis of Folland

I got stuck on this problem and couldn't find any clue to solve it. Can anyone give me some hint or give me some solution for it. I really appreciate! Suppose that $\mu(X) < \infty$ and ...
1
vote
1answer
43 views

Are infinite-dimensional singletons measurable?

Consider the wiener measure space $C[a,b]$ of all real-valued continuous functions on $[a,b]$ with the wiener measure $\mu$ on the $\sigma$-algebra $\mathcal{A}$ of Carathéodory measurable sets in ...
2
votes
2answers
70 views

Show $\int_E {(f_1 + f_2)d\mu } = \int_E {f_1 d\mu } + \int_E {f_2 d\mu } $

In my textbook, given a measure space $(\Omega,F,\mu)$, the integration for a non-negative $F$ measurable function $f$ on $E$ is defined as $$\int_E f\ \mathsf d\mu = \sup_{0 \le h \le f} I_E\left( h ...