Questions relating to measures, measure spaces, Lebesgue integration and the like.
1
vote
0answers
23 views
$u$ Absolutely Continuous: $|u(B)|=0$ implies $|B|=0$?
Let $u:[0,1]\to\mathbb{R}$ be a absolutely continuous function. It is know that $u'(x)$ exist almost everywhere and $u'\in L^1(0,1)$. Let $A=\{s\in [0,1]:\ u'(s)\ \mbox{exist and}\ u'(s)\neq 0 \}$. ...
4
votes
1answer
71 views
Non-Borel subsets of [0,1] and a definition in an article
I have the following problem to answer: let $B\subseteq \mathbb{R}^{n}$ be a compact set, let $b\in B^{[0,1]}$ and let $\{\pi_{i};i\in [0,1]\}$ denote the canonical projections. Is the following a ...
0
votes
1answer
82 views
Integration of a $BV$ function with respect to a finite, signed Radon measure
Let $u,w\in BV(0,1)$ be given. Since we are in dimension 1 $u$ is continuous almost everywhere and has a representation $u^{l}$ and $u^{r}$ (left, right hand side continuous). Thus we can consider the ...
0
votes
3answers
31 views
Dirichlet function involved with integral
Let the Dirichlet function $D:[0,\pi] \rightarrow R$ be given by
$D(x):=\begin{cases} 0 &\text{if } x\in [0,\pi] \cap Q, \\{}\\ 1 &\text{ otherwise}.\end{cases}$
For the function ...
-4
votes
0answers
40 views
Prove that two finite Borel measures are equivalent.
Let $\mu$ and $\nu$ be finite Borel measures such that $\mu$((-$\infty$, x])= $\nu$((-$\infty$, x]) for all x $\in$ $\mathbb{R}$. Prove $\mu$=$\nu$.
5
votes
0answers
122 views
+50
Operator completly continuous
For $\lambda>0$, let $v(t)=\lambda \phi(t)$. Consider the BVP
consisting of the equation $$u'''=\lambda[f(t,[u-v]^*+\gamma)+M(t)] ,t\in (0,1)$$
and (BC):$u(0)=u'(p)=\int_q^1 w(s)u''(s)=0 ...
1
vote
0answers
25 views
Not every $1$-nullset is an $s$-nullset, for $0 < s < 1$
For any positive real number $s$, define an $s$-nullset as a subset $A$ of the real line such that, for any $\epsilon > 0$, there exists a sequence of intervals $\{I_n\}_{n=1}^{\infty}$ having the ...
0
votes
0answers
16 views
What is the measure of all probability distributions with finite variance?
I'm in over my head here, but I am wondering about the probability that a distribution has finite variance? (or a finite mean?)
By this, I don't mean that there is some set of data, just over the set ...
0
votes
1answer
23 views
Uniform convergence , absolute integral, and a.e. convergence
I found this on a qualifier exam, and I think it will help me understand convergence better.
$\{f_n\}$ are real-valued integrable functions on $I=[0,1]$, then consider below three statements:
(a). ...
1
vote
0answers
22 views
Jordan measure on the unit interval
It is known that Jordan measure is finitely additive measure and the set of all Jordan measurable forms a Boolean algebra $\mathcal{B}$. My question is ?
Show that the Jordan measure on $[0,1]$ ...
8
votes
0answers
122 views
+50
From universal measurability to measurability
Let $(\Omega,\Sigma)$ be a measurable space and $K$ be a compact
metrizable space endowed with its Borel $\sigma$-algebra
$\mathcal{B}(K)$. Let $A\subseteq\Omega\times K$ be universally
...
4
votes
1answer
60 views
Discontinuous Almost-Everywhere/ Unbounded in $L^{1}(\mathbb{R})$
Let $L^{1}(\mathbb{R})$ be defined as usual, with the equivalence relation : $f \approx g$ if and only if $f(x) = g(x)$ almost everywhere.
Is there a class in $L^{1}(\mathbb{R})$ such that every ...
0
votes
2answers
21 views
Prove $\exists\;E$ such that $\mu(E)=0$ and $\alpha=\sup\limits_{x\in X\backslash E} |f(x)|$
$(X,\mathscr{M},\mu)$ is a measurable space and $f$ is a measurable function on $X$. Denote by $$\alpha=\inf\{\sup\limits_{x\in X\backslash E} |f(x)| : E\in\mathscr{M}, \mu(E)=0\}.$$
Note that the ...
3
votes
1answer
38 views
Prove that $\lim\limits_{n\to\infty} \int\limits_X f_nd\mu=\int\limits_X fd\mu$
$(X,\mathscr{M},\mu)$ is a measurable space and $f_n,g_n,f,g$ are all measurable functions defined on $X$. The following conditions are satisfied:
(i) $f_n\to f\;\&\;g_n\to g$ almost everywhere ...
0
votes
2answers
23 views
Equal Measurable functions
I am trying to understand measure theory and I cannot figure out how to prove the following statement: if $g,h:[0,1]\rightarrow R$ are measurable functions, then the set {$x\in [0,1]:g(x)=h(x)$} is a ...
0
votes
1answer
23 views
Understanding Properties of Measure
I am trying to understand measure and was thinking about the following:
If X and Y are open subsets of R such that $X\subset Y$ but X and Y are not equal, then $m(X)<m(Y)$. Is this statement true? ...
1
vote
3answers
57 views
Showing Dirichlet function is measurable
Consider the Dirichlet function $D:[0,\pi] \rightarrow R$ given by
$D(x):=\begin{cases} 0 &\text{if } x\in [0,\pi] \cap Q, \\{}\\ 1 &\text{ otherwise}.\end{cases}$
Then give an argument for ...
1
vote
0answers
50 views
Calculate the following: $\lim\limits_{n\to \infty}\int_X n \log(1+(\frac{f}{n})^{\alpha})d\mu$
Let$(X,\mathcal{M},\mu)$ be measure space, $f$ is non-negative integrable function on $X$, and $\int_X fd\mu=c,0<c<\infty$.Then calculate the following:
$$\lim\limits_{n\to \infty}\int_X n ...
0
votes
0answers
48 views
Creating non-measurable functions from Cantor function
I encountered the following problem in which I don't understand the construction it is proposing:
The Cantor d.f. $F$ is a good building block of "pathological"
examples. For example, let ...
2
votes
1answer
28 views
measure of the image of the the unit open disc by a holomorphic map
I found the following interesting exercice in a textbook:
Let $f$: $\Bbb E \to \mathbb{C}$ be a holomorphic and injective map ('Schlicht function'), where $\Bbb E=\{z \in \Bbb C:|z|<1\}$. ...
2
votes
1answer
37 views
How uniform is the distribution of $n+sm$ for an irrational $s$?
It's not difficult to prove that for $s\in\mathbb{R}\setminus\mathbb{Q}$ the set $S=\{n+ms\;|\;n\in\mathbb{Z},\;m\in\mathbb{N}\}$ is dense in $\mathbb{R}$.
When trying to solve this question, I come ...
0
votes
1answer
36 views
Outer measure, sequence of subsets proof
Let $I_o=[a,b]$ be a fixed interval and let $A$ be a subset of $I_o$. Show that if $\{A_n\}^{\infty}_{n=1}$ is a sequence of subsets of $I_o$ such that $m^*(A_n)=0$ for every $n \in N$ then
...
0
votes
0answers
38 views
Show that a function $g$ is measurable and summable.
Given a function $g:[a,b]\to\mathbb{R}$ that is of bounded variation, show that $g$ is measureable and summable.
I am familiar with the definitions, but don't even know where to start the proof. Any ...
-2
votes
0answers
37 views
Show that $ \lim\limits_{y \to 0}\int_R |f(x+y)-f(x)|dx=0 $ for $f\in\mathcal{L}^1(R)$. [duplicate]
$f\in\mathcal{L}^1$
Show that
$ \lim\limits_{y \to 0}\int_R |f(x+y)-f(x)|dx=0 $
for $f\in\mathcal{L}^1(R)$.
Definition of $||f||_p:=(\int_R|f(x)|^pdx)^{1/p}$
And calculate $\lim\limits_{|y| \to ...
-2
votes
1answer
80 views
Showing that $\{f_n \}$ converges to $f$ is equivalent to $\lim_{n\to \infty} \int_X \frac{|f_n(x)-f(x)|}{1+|f_n(x)-f(x)|}d\mu(x)=0$
Let $(X,\mathcal{M},\mu)$ be a measure space. We say that $\{f_n\}$ converges to $f$ in measure if, for any $\epsilon>0$
$$ \lim\limits_{n \to \infty} \mu\Big(\{x\in X: |f_n(x)-f(x)| \ge ...
-4
votes
2answers
56 views
Suppose $f,g$ are measurable functions on $X$, then show the set $\{ x\in X:f(x) >g(x) \}$ is $\mu$-measurable.
(1) Suppose $f,g$ are measurable functions on $X$, $\rightarrow$ need to prove
the set $\{ x\in X:f(x) >g(x) \}$ is $\mu$-measurable.
(2) Suppose $f$ is positive measurable function on $X$,and ...
0
votes
1answer
28 views
Need an Explainantion
I hope someone could spend some seconds to help me understanding this paragraph.
The paper by Von Neumann:
This part is little bit confusing for me
Let $E$ be a set of positive measure which ...
3
votes
1answer
62 views
Existence of a probability measure under which a subset of $[0,1]$ cannot be approximated by a Borel set
Is there some probability measure, $p$, on $\left(\left[0,1\right],2^{\left[0,1\right]}\right)$ relative to which some $D\subseteq\left[0,1\right]$ cannot be approximated by a Borel set, i.e. if $E$ ...
-4
votes
0answers
31 views
Measure space $(X,\mathcal{B},\mu)$ ,$\mu(X)< \infty$, $f$ is measurable function on $X$, $||f||_{\infty}:=ess. sup |f(x)| < \infty$.
I. Measure space $(X,\mathcal{B},\mu)$ ,$\mu(X)< \infty$, $f$ is measurable function on $X$, $||f||_{\infty}:=ess. sup |f(x)| < \infty$.
I need to solve the below.
(1)Show $\mu(\{x\in X| ...
-1
votes
0answers
63 views
Problem 25 pg 95, Stein and Shakarchi: $F(\xi) = 1/(1+|\xi|^2)^\epsilon$ is the Fourier transform of a function in $L^1(R^d,m)$.
Show that for any $\epsilon>0$, the function $F(\xi) = 1/(1+|\xi|^2)^\epsilon$
is the Fourier transform of a function in $L^1(R^d,m)$.
[Hint: $K_{\delta}(X) = e^{-\pi|x|^{2/\delta}} ...
1
vote
2answers
75 views
Problem #23 pg-94, Stein and Shakarchi
As an application of the Fourier transform, show that
there does not exist a function $I\in L^1(R^d,m)$ such that
$f*I = f$ for all $f\in L^1(R^d,m)$.
1
vote
0answers
70 views
Problem # 25, page 95, from Stein and Rami [duplicate]
Let $(X,M,\mu)$ be a measure space with $\mu(X) < 1$. Show that for any $1\le p<q$, we have $$L^q (X,\mu)\subset L^p(X,\mu).$$ Let $\ell^p(Z)$ denote the $L^p$ space of the integers equipped ...
4
votes
1answer
34 views
Prove that it is a random variable iff it is constant on each partition
Let $\mathcal{G} = \{A_1, \ldots, A_n\}$ be a partition of a set $\Omega$, $\mathcal{F} = \sigma(\mathcal{G})$. Prove that $X : \Omega\to\mathbb{R}$ is a random variable if and only if it is constant ...
0
votes
1answer
32 views
Measurability w.r.t. the $\sigma$-field induced on the range II
I'm sorry, but i asked the wrong question here. What i meant to ask was whether $f^{-1}\left(\mathcal{T}\right)=f^{-1}\left(\mathcal{T}'\right)$ in case $\left(T,\mathcal{T}\right)$ is a Borel space.
0
votes
1answer
30 views
Measurability w.r.t. the $\sigma$-field induced on the range
I asked here the wrong question. See here for the correct one.
Let there be given two measurable spaces $\left(S,\mathcal{S}\right)$ and $\left(T,\mathcal{T}\right)$ and suppose $f$ is a mapping ...
2
votes
1answer
36 views
Proof of conditional expectation
Suppose that we have three integrable random variables $x,y,z$ on a probability space $(X,\Sigma, \mathbb{P})$ such that $x$ and $z$ are independent, and $y$ and $z$ are also independent. Show that ...
4
votes
0answers
48 views
Conditional expectation as a random variable
We have three random variables $x,y,z$. Is the condition "$y$ and $z$ are independent" enough to guarantee that "$\mathbb{E}(x\,|\,y)$ and $z$ are independent"? Would anyone give me a brief proof or ...
2
votes
1answer
34 views
About independence and conditional expectation
Can anyone give me a little hint on a the following question? Many thanks!!
The question is: If we know that $x$ and $z$ are independent, and $y$ and $z$ are independent, is it true that
...
0
votes
0answers
31 views
Tail $\sigma$-algebra
With a simple symmetric random walk such that $S_n=\sum\limits_{k=1}^n X_k$ and $\mathbb{P}[X_i=\pm1]=1/2$ with $S_0=0$ like in this post: Tail events and exchangeable events where Did answered some ...
1
vote
2answers
51 views
About conditional expectation
Can someone give me some hints on the following problem? Many thanks!!
Let $x$, $y$, and $z$ be integrable random variables on a probability space $(X,\Sigma, \mathbb{P})$. Show that if both $x$ and ...
4
votes
2answers
104 views
Minimizing the expectation over a set, wrt to the Gaussian measure
I have recently read a proof [1] where, at the last step, the authors use an inequality which basically amounts to a lower bound on
$\int_\mathbb{R} \mathbf{1}_A(x)|x| \phi(x)dx$, where $\phi$ is the ...
0
votes
1answer
57 views
Computing outer measure
Compute $m^*(\{(1+\frac{1}{n})^n:n\in N\})$
I'm fairly new to outer measures and having trouble using the definition of an outer measure to compute this. Thank you for any help!
0
votes
1answer
18 views
If $P$ is a statistically complete set of distributions, the only sufficient subfield is the trivial one
In this thread i solved a claim stated without proof by Bahadur that if $P=\left\{p\right\}$ is the set of all probability measures on the measurable space $\left(\Omega,\mathcal{A}\right)$, ...
0
votes
1answer
19 views
Outer Measure of a Finite Covering of the Rationals on $[0, 1]$
I'm studying for my Real Analysis final and came upon an old question on outer measure that I'm pretty sure I'm doing wrong.
If $B$ is the set containing the rationals on $[0, 1]$, and ...
1
vote
1answer
38 views
Tail events and exchangeable events
In this problem I have $X_1, X_2, \cdots$ independent identically distributed RVs taking values $\pm1$ with the equal probability of $1/2$ and my trajectory is defined by $S_n=\sum_i^n X_i$ (so pretty ...
2
votes
1answer
31 views
Can I deal with the weak derivative in the “strong” sense?
This is an exercise in functional analysis:
For $k=1,2,3$, let $A_k: D(A_k)\subset L^2([0,1])\to L^2[(0,1)]$ be the first-order differential operators $A_ku=iu'$ with domains
$$
D(A_1) = ...
2
votes
2answers
55 views
Why this is not a sigma algebra
Let be $\Omega$ the interval $(0,1]$ and let $\mathcal{F}$ be the set of all sets of the form $(a_0,a_1]\cup(a_2,a_3]\cup\cdots\cup(a_{n-1},a_n]$, where $0\le a_0\le a_1\le\cdots\le a_n\le 1$. Show ...
1
vote
2answers
49 views
Question in Lebesgue integrable functions.
Suppose $g$ be a measurable function satisfying: $∀$ $σ∈[c,d]$ , there exists $δ>0$ such that $∫_E|g| <∞$ where $E=[σ-δ, σ+δ]$. Prove that $g$ is Lebesgue integrable on $[c, d]$.
3
votes
0answers
48 views
A question about the stability of a property of the normal distribution
Recall that the standard normal distribution can be characterized as the unique standardized (having mean zero and unit variance) distribution $P$ on $\mathbb{R}$ with the property that with $X$, $Y$ ...
0
votes
1answer
47 views
Monotone increasing sequence of random variable that converge in probability implies convergence almost surely
Let $\{X_n\}$ be a collection of random variable with $X_{n+1} \geq X_n$ for all $n$ and $X_n \rightarrow X$ in probabilty. How to prove that $X_n \rightarrow X$ almost surely.
My partial answer:
...
