Questions relating to measures, measure spaces, Lebesgue integration and the like.
0
votes
0answers
28 views
Gap distribution independence proof
I have a question bout the proof of the independence of gap RVs. Given the independent exponentially distributed random variables $\xi_1$, $\xi_2$ ~ $\text{Exp}(\lambda)$, and a corresponding order ...
2
votes
1answer
85 views
A representation theorem for a minimally sufficient statistic by Bahadur
The Statement of the Problem
I'd appreciate help in proving the following, unproven theorem from a classic article by Bahadur ([BAH], Theorem 6.3) (the expressions in square brackets are my ...
0
votes
0answers
43 views
Sets of Measure Zero
i would like to understand correctly what does mean set of measure of zero?for example in my book there is written statement something like this:
suppose we have continuous, monotone bounded ...
0
votes
1answer
37 views
Radon-Nikodým derivative with respect to the Lebesgue/Hausdorff measure is always defined
Question 1. Is it possible to say that the Radon-Nikodým derivative of locally-finite Borel measure on $\mathbb R^n$ with respect to the Lebesgue measure is always defined but may be a generalized ...
1
vote
0answers
28 views
Cylindrical sigma algebra answers countable questions only.
I got a missing link in some in the following (standard) textbook question:
Show that the cylindrical sigma algebra $\mathcal{F}_T$ on $\mathbb{R}^T$ (equals $\bigotimes_{t\in ...
3
votes
1answer
43 views
How to show that this functional is lower semicontinuous?
The functional is given by:
$J(y) = \int_{a}^{b} \sqrt{1 + y'(x)^2} dx$
and I need to prove that it is lower semi-continuous with respect the norm:
$|| y || = \max_{a \leq x \leq b} |y(x)|$
and $y ...
0
votes
1answer
57 views
Measurable function is bounded almost everywhere
Let $f : [a, b] \to \mathbb{R}$ be a measurable function. Given $\varepsilon > 0$ show that there is some $M > 0$ such that $|f(x)| \leq M$ for all $x \in [a, b]$ except on a set of finite ...
0
votes
0answers
40 views
Step function existence for a simple function
Suppose that $\varphi : [a, b] \to \mathbb{R}$ is a simple function and let $\varepsilon > 0$ be given. Prove that there is a step function $g : [a, b] \to \mathbb{R}$ such that $g(x) = \varphi(x)$ ...
3
votes
0answers
30 views
Existance of a simple function
Let $f : [a, b] \to \mathbb{R}$ be a measurable function. Suppose that $\varepsilon, M > 0$ are given. Show that there is some simple function $\varphi : [a, b] \to \mathbb{R}$ such that $|f(x) - ...
3
votes
1answer
55 views
Bounded variation implies Borel measurable
Suppose that $f\colon[a, b] \to \mathbb{R}$ is a function of bounded variation. Show that $f$ is Borel measurable.
I was wondering if I could get a hint.
1
vote
1answer
18 views
Measurable Set From Cauchy sequences
Suppose that $D$ is a measurable set and that for each integer $n \geq 1$, $f_n : D \to \mathbb{R}$ is a measurable function. Prove the set
$$E = \{x \in D \mid (f_n(x))_{n \geq 1} \text{ ...
6
votes
1answer
42 views
How can a $\sigma$-algebra be “treated” or computed? Example
My question is: I have a random variable $X:\Omega \rightarrow \mathbb{R}$, the $\sigma$-algebra generated by $X$ is: $\sigma(X) := \{X^{-1}(B), B\in \mathcal{B}(\mathbb{R})\}$.
But, imagine now that ...
0
votes
1answer
31 views
Polar form for $f\in L^2(\mathbb{R}^n;\mathbb{C})$
I have some doubts in measure theory. Suppose $f\in L^2(\mathbb{R}^n;\mathbb{C})$, then $f=f_1+if_2$, where $f_1,~f_2\in L^2(\mathbb{R}^n;\mathbb{R})$. Is it possible to write this function in a polar ...
2
votes
0answers
34 views
showing to be extreme subset (might use Hahn decomposition Theorem)
I am studying Functional analysis by myself and stumbled this question and am completely at a loss.
We want to show that $\{ f \in L^1 [0,1 ] : ||f|| =1 \}$ is an extreme subset of $\{ \mu \in ...
2
votes
2answers
44 views
Proof concerning outer measure
Assume that $X:=[a,b]$ is a fixed interval in $\mathbb R$ and let $m^*$ be the outer measure on $X$. Suppose that $A \subset X$ is a null set, i.e. $m^*(A)=0.$ Show that for every $B\subset X$,
...
1
vote
1answer
35 views
Properties of Lebesgue functions
If $f\in \mathcal {L}$ then there exists a sequence $\{f_k\}$ of step functions s.t. $\lim_{k\to\infty} f_k(x)=f(x)$ for almost all $x$ and $$\lim_{k\to\infty} \int|f(x)-f_k(x)|\,dx=0.$$ If I have ...
3
votes
2answers
76 views
Convergence of random variables (Durrett: Probability Theory and Examples)
I was working out some problems from Rick Durrett's Probability theory and Examples (2010 edition), when I came across a very unusual question(reproduced here ad-verbatim):
If $X_n$ is ANY sequence ...
0
votes
1answer
40 views
Outer measure defined with rectangles
I'm studying Measure Theory by myself and I would appreciate some guidance about my proof.
My textbook constructs an outer measure as following:
$$m_*(E)=\inf\sum_{k=1}^{\infty}|Q_k|$$
where the ...
0
votes
1answer
47 views
Outer measure, Caratheodory measure - proof
Let $m^*$ be an outer measure on a set $X:=[a, b]$. $A \subset X$ is a null set, i.e., $m^*(A) =0$. If $E \subset X$ is measurable, show that
$m^*(E\cup A)+m^*(E\cap A) = m^*(E)+m^*(A)$
I'm pretty ...
9
votes
0answers
88 views
Restrictions of null/meager ideal
Let I denote the null/meager ideal on reals. Is it consistent that for any pair of non null/meager sets A and B, there is a null/meager preserving bijection between A and B? In particular, is this ...
0
votes
0answers
22 views
Significant digits/rounding
I have a bunch of criteria to evaluate for a product, and each is scored on a scale from 0 to 5. Each criterion has a weight associated with it.
If I find the weighted score of a criterion, it is
...
1
vote
1answer
70 views
Continuous on rationals, discontinuous on irrationals
Let $f: R \rightarrow R$. Show that the set of points of continuity of
$f$ is a $G_{\delta}$ set. Explain why it follows from this that there
is no function that is continuous on the rationals ...
1
vote
3answers
80 views
Convergence in $L^1$
Let $f \in L^1(R)$. Show that $\sum^{\infty}_{n=1} f(x + n)$ converges
a.e.
Solution:
So, ultimately we are going to want $\sum^{\infty}_{n=1} f(x + n) \leq$ something in $L^1$ that converges ...
2
votes
0answers
45 views
Moment Generating Function Supremum
I'm studying probability, and am having trouble with the following:
Let $X$ be a uniform R.V. on $[0,1]$. Compute the moment generating function $M(t)$ of $X$. Compute $I(y) = \sup_{t \in ...
0
votes
0answers
30 views
Why is K-L divergence defined as it is?
Why is the K-L divergence defined this way:
if $P$ and $Q$ are probability measures over a set $X$, and $P$ is absolutely continuous with respect to $Q$, then the Kullback–Leibler divergence from ...
0
votes
1answer
64 views
Motivation for Measure Theory example
I was taking a look at this book while trying to pick a book for learning some rigorous probability theory. I have been totally stumped by the motivating eg. on the first page.
Specifically, I am ...
2
votes
1answer
53 views
Does an absolutely integrable function tend to $0$ as its argument tends to infinity?
Suppose that $f:[0,\infty)\rightarrow\mathbb{R}$ is continuous. Is it true that
$$\int_{0}^\infty|f(t)|dt<\infty\Rightarrow \lim_{t\rightarrow\infty}f(t)=0?$$
If so can you provide a proof, ...
1
vote
1answer
43 views
Equality of two limits of r.v.
considering a sequence of real-valued r.v. $(X_n)$ convergent to $X$ in probability. Moreover we look at a sqeuence of r.v. $(Y_n)$, where $Y_n\in\operatorname{conv}(X_n,X_{n+1},\dots)$ and we suppose ...
2
votes
1answer
35 views
Proving that $\sigma(\tau_{\mathbb{R}}\times\sigma(\tau_{\mathbb{R}}))=\sigma(\tau_{\mathbb{R}})\otimes \sigma(\tau_{\mathbb{R}})$
$\sigma(\tau_{\mathbb{R}})$ denotes the Borel $\sigma$-algebra ($\tau_{\mathbb{R}}$ is the usual topology on $\mathbb{R}$), $\sigma(\tau_{\mathbb{R}}\times\sigma(\tau_{\mathbb{R}}))$ is the ...
0
votes
2answers
44 views
If a continuous function from $\mathbb{R}$ to $[0,\infty)$ does not tend to zero, is its integral greater or equal than some linear function?
Consider a continuous function $f:\mathbb{R}\rightarrow[0,\infty)$ that does not tend to zero as its argument tends to infinity. Formally, there is some $\varepsilon>0$ such that there does not ...
4
votes
2answers
52 views
Find f ae-differentiable with $f´\in L^1(0,1)$ but not in $BV$…
Here is a natural question which I didn't find in Measure Theory books:
Construct a continuous function $f(x)$ in $[0,1]$ with derivative at ae $x\in(0,1)$, and so that $f'(x)\in L^1(0,1)$, but such ...
3
votes
2answers
158 views
An identity involving Radon-Nikodym derivatives
The following result was stated without proof in [HAL] (note 4 to section 32 "Derivatives of Signed Measures", p. 136).
If $\mu_0$, $\mu_1$, and $\mu_2$ are finite measures, and if
$$
...
0
votes
1answer
26 views
Difference in reference of L space in Fubini Tonelli
I think my understanding of how $L^+$ and $L^1$ spaces are defined (I'm using Folland) is a little hazy. For example, in the Fubini-Tonelli theorem:
$\textbf{For the Tonelli part:}$ We start with if ...
1
vote
0answers
35 views
Expectation of Random Variables - Measure Theory
I am trying to do the exercise 2 of section 3.2 of the book "A Course in Probability Theory by Kai Lai Chung". Problem asks to show:
If $\mathscr E(\left|X\right|)<\infty$ and $\lim_{n\to ...
2
votes
1answer
44 views
Outer Measure Question
Prove or give a counter example:
For every open set $U$ of $\mathbb{R}$, $m^*(\bar{U} - U ) = 0.$
My first impression was that it was true, since if $U$ is an open set in $\mathbb{R}$, then it can be ...
2
votes
1answer
33 views
Abstract integral - Borel measures - $L^p$ spaces
Let $(X,\mu,M)$ be a finite measure space. Suppose $T\colon X \to X$ is measurable and $\mu(T^{-1}E) = 0$ whenever $E \in M$ and $\mu(E)=0$. Prove that these exists $h \in L^1(\mu)$ such that $h ...
1
vote
2answers
86 views
Pairwise measurable derivatives imply measurability of combined derivative
I've found the following simple claim in an article. Unfortunately, i don't understand the proof given there nor can i come up with an alternative proof of my own. Maybe math.stackexchange can give me ...
4
votes
1answer
63 views
Is Cesaro convergence still weaker in measure?
I've encountered a question I couldn't answer, and I would appreciate any help:
Is it true that $f_n \xrightarrow{m}0$ $\Rightarrow$ $ \frac{1}{n} \sum_{k=1}^{n}f_k \xrightarrow{m}0$?
Where ...
1
vote
2answers
48 views
Every Lebesgue measurable function with bounded support is nearly bounded.
Let $f$ be a Lebesgue measurable function over the (non extended) reals with bounded support. I was wondering if we can say that, for every $\epsilon > 0$ there exists a bounded function $g$ such ...
0
votes
1answer
27 views
Integrable functions non-negative
Show that exist integrable functions non-negative that aren't equal in nearly all point the a function of type $\sum^{\infty}_{n=1}h_n$, $h_n$ $S$-simple.
(I'm sorry for english.)
0
votes
1answer
41 views
About equivalent characterization of ergodicity
Can anyone give me some hint on the following problem? Many thanks!
Given a probability space $(X, \Sigma, \mathbb{P})$ and a $\mathbb{P}$-preserving map $\tau: X\to X$, show that the following three ...
1
vote
1answer
14 views
Open bounded set $E$ so that $m(E)\neq\lim_{n\rightarrow \infty}m(O_n)$
Let $E$ be a compact set and let us define the series:
$$O_n=\{x\in R^d |d(x,E)<1/n\}$$
I proved that:
$$m(E)=\lim_{n\rightarrow \infty}m(O_n)$$
Now I'm trying to find an open bounded set $E$ for ...
3
votes
2answers
58 views
Regular Borel Measures equivalent definition
Please help me understand how the below definition is equivalent to the standard definition of regularity which says "that a measure is regular if for which every measurable set can be approximated ...
1
vote
1answer
24 views
Show that $B(R^d)$ is the smallest $\sigma$-algebra satisfy the condition
Show that collection of all Borel set in $R^d$ (i.e. $B(R^d)$) is the smallest $\sigma$-algebra which make all continuous functions on $R^d$ measurable
0
votes
1answer
67 views
Prove $A=\{x\in X:f(x)=g(x)\}\in\mathcal F$
Let $R_0=[-\infty,\infty]$ and $(X,\mathcal F)$ be a measurable space.
If $f,g:X\to R_0$ is $\mathcal F-B(R_0)-$measurable functions then prove that $A=\{x\in X:f(x)=g(x)\}\in\mathcal F$.
2
votes
0answers
38 views
How to determine a measure out of a positive linear functional
let $X=[-2,2]$. $l$ is a positive linear functional on $C([-2,2])$ such that $l(x^{2n})=C_{2n+2}^{n+1}$ and $l(x^{2n+1})=0$ can we determine the measure corresponding to this positive linear ...
0
votes
0answers
45 views
Lebesgue integral of a bounded measurable function over a measurable subset of a measurable set of finite measure.
Let $f$ be a bounded measurable function on a set $E$ of finite measure. For a measurable subset $A$ of $E$, show that $\int_A f=\int_E f\cdot\chi_A$, where $\chi_A$ is a characteristic function on ...
1
vote
3answers
25 views
Simple Function attains a maximum?
Let $A_1,..,A_n \subset \mathbb{R}^n$ be sets with finite measure and let $a_1,\ldots,a_n$ be real numbers. Consider the simple function $$f(x)=\sum_{k=1}^n a_k \chi_{A_k}$$
where $\chi_A$ is the ...
0
votes
0answers
52 views
Change of differentation and integration signs.
I'm going through an old exam in a course I'm taking. I have the given rule:
Let $X$ be a measure space, $U$ be open subset in $\mathbf{C}$ and $f: U \times X \to \mathbf{C} $ be a function s.t. the ...
1
vote
1answer
38 views
Markov/Chebyshev Inequality
I am looking at proofs of Markov or Chebyshev's inequality that for a measurable function, the set $B=\{x\in\mathbb R^n:|f(x)|\ge t\}$ where $0\lt t\lt \infty$ , has a measure that is ...
