Questions relating to measures, measure spaces, Lebesgue integration and the like.
3
votes
1answer
42 views
A (partial) argument converting sums in $\ell^1$ into Lebesgue-integrable functions.
First, I want to mention that this problem is off of a take home final I have. I was given permission to research/ask about this specific line of reasoning, in large because I think the professor ...
2
votes
2answers
54 views
Any example for a function having domain and range as subset of real line that is NOT Borel function?
Suppose there is a function $f:A\to B$ where $A,\,B\subseteq\mathbb{R}$, is there any example for this function being NOT Borel function?
Well the question came up to be when I was reading the ...
0
votes
2answers
65 views
Outer measure proof, assuming measurable set exists
Let $A\subset X $ be a null set (so $m^*(A)=0$). Assume that $X:=[a,b]$ is a fixed interval in $\mathbb R$ and let $m^*$ be the outer measure of $X$. Show that $A\subset X$ is a measurable set if and ...
2
votes
1answer
51 views
Interchange differential operator with Lebesgue integral.
Under what condition am I able to interchange a differential operator with an integral? More precisely, given a function $f:\Omega\times U\to\Bbb R$ from a measure space $(\Omega,\mathscr A,\mu)$ and ...
0
votes
2answers
40 views
densities being absolutely continuous wrt Lebesgue measure
I'm reading an article with an assumption similar to: "The density $f(.)$ exists and is absolutely continuous with respect to Lebesgue measure". I don't understand this assumption because $f$ is not ...
4
votes
0answers
104 views
strict convexity with a measure theoretic property
Suppose $(x_n)$ is a positive sequence of reals converging to $x$. Furthermore we have a measure space $(E,\mathcal{E},\mu)$ given, with finite measure $\mu$. There are a measurables nonnegative and ...
0
votes
1answer
56 views
Probability of two events are indepedent
Given a probability space $(\Omega, \mathscr {B}, P)$, then $\sigma : \mathscr{B}\times \mathscr{B} \to [0,1]^2$ is defined as, for any $A, B \in \mathscr{B}$ $$(A,B) \mapsto (P(A),P(B))$$
Now take ...
5
votes
3answers
144 views
What measure does Lebesgue measure induce on the fat Cantor set?
I know that the fat Cantor set under the subspace topology is homeomorphic to Cantor space $\{0,1\}^{\mathbb N}$ under the product topology induced by the discrete topology on $\{0,1\}$. Call the ...
6
votes
1answer
91 views
Image of a set of zero measure has zero measure
I am studying for my final and got stuck on the following problem from the previous year. I put my attempt below.
Suppose that $I\subset \mathbb{R}$ is an open interval, $f:I\rightarrow \mathbb{R}$ ...
1
vote
1answer
39 views
Extension of Fourier Transform
We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ ...
0
votes
2answers
55 views
Strict inequality in Reverse Fatou lemma: $\varlimsup \int f_n\le \int \varlimsup f_n$
Let $\{f_n\}$ be a sequence of nonnegative functions dominated by some function
$$
g \in L^1.
$$
Then, the reverse Fatou lemma says
$$
\limsup \int f_n \le \int \limsup f_n.
$$
Is it possible to ...
5
votes
1answer
150 views
Interchange supremum and expectation
Let $B_n:=\{f\in L^\infty_+\mid f\le n \}$, where we consider $L^\infty$ with the weak$^*$ topology. I have the following sets
$$D(z):=\{h\in L^0_+(\mathcal{F}_T)\mid h\le Z_T \mbox{ for a }Z\in ...
0
votes
1answer
55 views
Counterexamples for Borel-Cantelli
Our teacher mentioned to construct two counterexmaples for Borel-Cantelli using the following ways.
(a) Construct an exmaple with $\sum_{i=1}^{\infty}\mathbb P(A_i)=\infty$ where $\mathbb ...
2
votes
1answer
32 views
Lebesgue Integral defined on infinite measure
Royden's Real Analysis Question: Let {$a_n$} be a sequence of nonnegative real numbers. Define the function $f$ on $E=[1,\infty)$ by setting $f(x)=a_n$ if $n\leq x< n+1$. I want to show that ...
0
votes
1answer
28 views
Computing $\mathbb{E}[X_1S^4]$
Given $X_1,X_2,\cdots$ i.i.d. random variables with $\mathbb{E}[X_i]=0$. If we are given $S=\sum_{i=1}^{10}X_i$ and the fact that $\mathbb{E}[S^5]=30$ what method do you need to compute ...
2
votes
1answer
53 views
Verifying Fatou's Lemma
Royden's Real Analysis Question: Let {$f_n$} be a sequence of nonnegative measurable functions on $R$ such that $f_n\implies f$ pointwise on $E$. Let $M\geq0$ be such that $\int_Ef_n\leq M$ for all ...
1
vote
0answers
42 views
Showing that a piecewise function is measurable
I'm doing an assignment for a (first) course in analysis, and I'm having some trouble with showing that functions are measurable.
In this problem $(X,\mathcal{A},\mu)$
and ...
0
votes
0answers
28 views
Exponential Order Statistics Independence
Are the order statistics from the $n$-sample with $X_i\sim \text{Exp}(\lambda)$ (taking, without loss of generality, $\lambda=1$) $\Delta_{(k)}X=X_{(k)}-X_{(k-1)}$ independent?
Can show that for an ...
3
votes
2answers
52 views
Intersection of $\sigma$-algebras and set theory
Theorem: Given $\{E_{\alpha}\}_{\alpha \in \mathcal{A}}$, where each $E_\alpha$ is a $\sigma$-algebra on $X$. Then $E:=\bigcap_{\alpha \in \mathcal{A}}E_\alpha$ is a $\sigma$-algebra.
Proof: Take ...
3
votes
1answer
32 views
Application of Strong Law of Large Numbers and Fubini's Theorem
This problem comes from here. I am not looking for help on solving the problem, actually to understand something said in the setup:
Let $F$ be a distribution with $F(0-) =0 $ and$F(1)=1$. Let ...
0
votes
1answer
63 views
Measure theory properties proof
For a set $A \subset \mathbb R$, $\alpha \in \mathbb R$ and $x_o \in \mathbb R$, put
$x_0 + A:=${$x_o+a:a \in A$} and $\alpha A:=${$\alpha a:a \in A$}
Let $m^*$ be an outer measure on a set ...
0
votes
1answer
39 views
Absolute Convergence of a Function
I have got stuck with a question. Please help me.
Prove that $\dfrac{\sin(x)}{x}$ belongs to $L^p$ for all $p>1$.
Thank You.
1
vote
0answers
75 views
Are continuous functions strongly measurable?
Measure theory is still quite new to me, and I'm a bit confused about the following.
Suppose we have a continuous function $f: I \rightarrow X$, where $I \subset \mathbb{R}$ is a closed interval and ...
1
vote
0answers
25 views
Borel measure supported on some set
Reading an article, I found the sentence : Let $\nu$ be a Borel measure on $(0,1)$ supported on $C$; $C$ is a compact totally disconnected subset of $(0,1)$, but it probably doesn't matter.
It is ...
1
vote
0answers
31 views
finitely additive measure “extension”
Suppose that a Boolean algebra $\mathcal B$ admits a finitely additive measure $\mu$ and $\mathcal A$ is dense in $\mathcal B$. Let $\mathcal{\bar{B}}$ is the completion of $\mathcal B$ and ...
2
votes
1answer
52 views
Concerning the noncompactness of the map, $I: W^{1, 2}(\mathbb{R}^n)\rightarrow L^{2^{\ast}}(\mathbb{R}^n)$
Consider the identity map $I:W^{1,2}(\mathbb{R^n})\rightarrow L^{2^{\ast}}(\mathbb{R}^n)$ where $n\geq 3$. Suppose that this map is not compact that is given some bounded sequence of functions ...
2
votes
2answers
109 views
Interpretation of dP in Radon-Nikodym Theorem
[Radon-Nikodym Theorem]
Let $(\Omega, \Sigma, P)$ be a probability space. Suppose that $(\Omega, \Sigma, \mu)$ be a measure space with $\mu(A)=0$ implies $P(A)=0$, then there exist a function $f:X ...
1
vote
0answers
83 views
Change of probability measure and a continuous-time Markov chain
Let $(\Omega,\mathcal{F},\mathbb{P},\mathbb{F})$ be a complete filtered probability space, with $W$ a Wiener process and $\alpha$ a continuous-time Markov chain (taking values in $\{1,...,M\}$). We ...
0
votes
2answers
62 views
Conditional Expectation of Exponential Order Statistic $\text{E}(X_{(2)} \mid X_{(1)}=r_1)$
Having already worked out the distributions of $\Delta_{(2)}X=X_{(2)}-X_{(1)}\sim\text{Exp}(\lambda)$ and of $\Delta_{(1)}X=X_{(1)}\sim\text{Exp}(2\lambda)$ where $X_{(i)}$ are the $i$th order ...
0
votes
1answer
27 views
A Borel measurable function which is not continuous
I want to find a example of Borel measurable function which is not continuous.
I think that it is a simple or step function or semicontinuous function.
Please help me for find it.
Thanks.
2
votes
0answers
61 views
When $ A \int_0^{\infty} e^{-\lambda t}S(t)u dt = \int_0^{\infty} e^{-\lambda t}S(t)Au dt$?
I have real Banach space $X$ and a bounded linear operator $S: X \to X$ which satisfy:
1) $S(0)u = u$ $\text{ }$ for all $u \in X$
2) $S(t+s)=S(t)S(s)u = S(s)S(t)u $ $\quad$($ t,s \geq 0$, $u \in X ...
1
vote
0answers
38 views
Does convergence in $L^p$ and pointwise imply the same limit?
If $f_n\in L^p$ converge to $f$ pointwise and to $g$ in $L^p$. Does that mean $f=g$ almost everywhere?
3
votes
1answer
80 views
Exponential Distribution Function
If $X\sim \text{Exp}(X)$ then for all positive $a$ and $b$, $P(X>a+b\mid X>a)=P(X>b).$ So given independent random variables $X \sim \text{Exp}(\lambda)$, $Y \sim\text{Exp}(\mu)$ we would ...
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
30 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
58 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
35 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
36 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
80 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 ...
