Questions relating to measures, measure spaces, Lebesgue integration and the like.
1
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0answers
13 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
30 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
1answer
21 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 ...
2
votes
2answers
54 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
22 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
29 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
29 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
29 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
69 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
54 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
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2answers
21 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
24 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.)
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0answers
20 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
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1answer
13 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)$$
Now I'm trying to find an open bounded set $E$ for ...
3
votes
2answers
48 views
Regular Borel Measures equivalent definition
Please help me understand how the below definition is equivalent to the standard definition of regularity which says "that a measure is regular if for which every measurable set can be approximated ...
1
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1answer
20 views
Show that $B(R^d)$ is the smallest $\sigma$-algebra satisfy the condition
Show that collection of all Borel set in $R^d$ (i.e. $B(R^d)$) is the smallest $\sigma$-algebra which make all continuous functions on $R^d$ measurable
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1answer
65 views
Prove $A=\{x\in X:f(x)=g(x)\}\in\mathcal F$
Let $R_0=[-\infty,\infty]$ and $(X,\mathcal F)$ be a measurable space.
If $f,g:X\to R_0$ is $\mathcal F-B(R_0)-$measurable functions then prove that $A=\{x\in X:f(x)=g(x)\}\in\mathcal F$.
2
votes
0answers
30 views
How to determine a measure out of a positive linear functional
let $X=[-2,2]$. $l$ is a positive linear functional on $C([-2,2])$ such that $l(x^{2n})=C_{2n+2}^{n+1}$ and $l(x^{2n+1})=0$ can we determine the measure corresponding to this positive linear ...
0
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0answers
28 views
Lebesgue integral of a bounded measurable function over a measurable subset of a measurable set of finite measure.
Let $f$ be a bounded measurable function on a set $E$ of finite measure. For a measurable subset $A$ of $E$, show that $\int_A f=\int_E f\cdot\chi_A$, where $\chi_A$ is a characteristic function on ...
1
vote
3answers
21 views
Simple Function attains a maximum?
Let $A_1,..,A_n \subset \mathbb{R}^n$ be sets with finite measure and let $a_1,\ldots,a_n$ be real numbers. Consider the simple function $$f(x)=\sum_{k=1}^n a_k \chi_{A_k}$$
where $\chi_A$ is the ...
0
votes
0answers
42 views
Change of differentation and integration signs.
I'm going through an old exam in a course I'm taking. I have the given rule:
Let $X$ be a measure space, $U$ be open subset in $\mathbf{C}$ and $f: U \times X \to \mathbf{C} $ be a function s.t. the ...
1
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1answer
26 views
Markov/Chebyshev Inequality
I am looking at proofs of Markov or Chebyshev's inequality that for a measurable function, the set $B=\{x\in\mathbb R^n:|f(x)|\ge t\}$ where $0\lt t\lt \infty$ , has a measure that is ...
0
votes
0answers
21 views
When a family of measures provide continuity?
Consider a mapping $m: \mathcal{B}(X) \times P \rightarrow [0,1]$, where $X \subseteq \mathbb{R}^n$, $P \subseteq \mathbb{R}^m$, and $\mathcal{B}(X)$ denotes the Borel sets.
$\forall p \in P$, ...
5
votes
2answers
59 views
Does $f_n \to 0$, a.e., implies $\int_{\mathbb R} \sin(f_n(x)) dx \to 0$, when each $f_n \in L^1$
Let $\{f_n\}$ be a sequence of $L^1(\mathbb R)$ functions converging a.e. to zero.
Does
$$
\lim_{n\to \infty} \int_{\mathbb R} \sin(f_n(x)) dx = 0?
$$
I think the answer is no, but I can't find a ...
1
vote
1answer
50 views
Prove there is a Borel measure u such that $u[x,y) = a(y) - a(x)$
If anyone has a solution to the following exercise, I would be grateful. Thanks.
Let $\alpha$ be continuous and increasing on $[a,b]$. Prove that there exists a unique Borel measure $\mu$ on ...
0
votes
1answer
35 views
Prove a function $f_\sigma\in C^\infty(\mathbb{R})$
Define $C^\infty(\mathbb{R})$ be the space that contains all bounded continuous functions on $\mathbb{R}$ that has continuous derivatives of all orders. Suppose $K\in C^\infty(\mathbb{R})$ s.t. ...
1
vote
3answers
67 views
A condition that balls have finite measure
Let $(X,d)$ be a metric space and let $\mu$ be a positive measure on $X$. I want to require that $(X,d)$ and $\mu$ have either of the following properties:
$\forall y \in X$, $\forall r \geq 0$, ...
1
vote
1answer
17 views
If a function is $L^p$ small, is its expectation with respect to a $\sigma$-algebra $L^p$ small?
This came up in my homework, but isn't strictly my homework. I've just gotten very curious, and I keep going in circles trying to prove it.
Consider a probability measure space $(X,\Sigma,\mu)$ and ...
3
votes
1answer
56 views
Random Walk on Z
Let $S_n$ be the symmetric random walk on $\mathbb{Z}$. How do i calculate
$P(\limsup_{n\rightarrow\infty} S_n=\infty)$?
I already know that the probability is 1 but I don't really know how to start? ...
1
vote
1answer
21 views
Inequalities for bounded lipschitz functions
Suppose $X$ is a metric space with metric $d$. Define $\lVert f(x)\rVert_\infty = \sup_x |f(x)|$ and $\lVert f(x)\rVert_{LIP}=\sup\{\frac{|f(x)-f(y)|}{d(x,y}:x\not=y\}$, let $\lVert ...
2
votes
1answer
47 views
Smallest $\sigma$-algebra generated by $\mathcal C$?
Can someone explain how this $\sigma$-algebra is attained? It's mainly the $X\cup Y$ bit which I don't understand.
Question: If
$\Omega = \{1, 2, 3, 4\}$ and we have a collection of sets
$\mathcal C ...
1
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0answers
44 views
Part of proof 11.10 in Rudin's Principles of Mathematical Analysis
There is a part of proof 11.10 that I don't get in Rudin's Principles of Mathematical Analysis (3rd edition).
The whole theorem is the following two statements:
$\mathcal{M}\left(\mu\right)$ is a ...
1
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2answers
28 views
Is $\frac{|C\cap B_r(x)|}{|B_r(x)|}$ decreasing in $r$?
Suppose $C$ is a measurable set, $x\in C$, is
$$
\frac{|C\cap B_r(x)|}{|B_r(x)|}
$$
decreasing in $r$? Or any counterexamples?
Thanks!
Edit:
@user39992 and @Karolis Juodelė show that it can not ...
2
votes
1answer
53 views
$\sigma$-algebras and independent stochastic processes
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space. We consider a Wiener process $W$ with respect to his standard filtration $(\mathcal{F}_t^W)_{t \geq 0}$ and a process $X$ with ...
0
votes
0answers
48 views
Riesz Representation Theorem and Indicator Function
I've been dealing with the Riesz Representation Theorem for measures and it is obvious that having a measure $\mu$ I can get a continuous linear functional $\mu^*$ in $C(X)^*$ where $X$ is a compact ...
1
vote
0answers
23 views
Is inverse function from metric space to pseudometric space borel?
Let $X$ be a compact metric space and $Y$ a pseudometric space, $f:X\rightarrow Y$ is continuous and bijective.
If there a non-trivial condition that makes $f^{-1}$ Borel?
As Martin commented it is ...
2
votes
0answers
40 views
Why does this inequality for all characteristic functions imply it for simple functions?
This question is probably obvious, but I'm not seeing how to obtain it.
A simple function is said to be finitely simple if its support is of finite measure. Let $(X_1,\mu_1)$, $(X_2,\mu_2)$, and ...
1
vote
1answer
29 views
$L^p$ convergence proof check
I don't have much experience with measure theory, so I want to make sure that I'm not making any bad mistakes. I also want to be sure that the theorem is true so I can use it.
Theorem: Let $\{u_i\}$ ...
5
votes
1answer
44 views
If $B\times \{0\}$ is a Borel set in the plane, then $B$ is a Borel set in $\mathbb{R}$.
I'm trying to figure out how to prove the following "obvious" fact:
Let $B\times \{0\}\subset \mathbb{R}^2$ be a Borel set, then $B\subset \mathbb{R}$ is a Borel set.
The problem here is that I ...
2
votes
1answer
40 views
A question about the proof of Rademacher theorem
I'm referring to the proof of Rademacher theorem due to C.B.Morrey (i'm reading it on Simon: 'Lectures on geometric measure theory').\
The proof can be summarized in the following steps:\
1)For every ...
5
votes
1answer
82 views
Conditional expectation on more than one sigma-algebra
I'm facing the following issue. Let $X$ be an integrable random variable on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $\mathcal{G},\mathcal{H} \subseteq \mathcal{F}$ be two ...
1
vote
2answers
49 views
Product sigma algebra of Borel sigma algebra and Power set.
Let $([0,1],\mathcal{B},m)$ be the Borel sigma algebra with lebesgue measure and $([0,1],\mathcal{P},\mu)$ be the power set with counting measure. Consider the product $\sigma$-algebra on $[0,1]^2$ ...
1
vote
1answer
42 views
$f_n\to f $ in $L^1$ $\implies$ $\sqrt{f_n}\to\sqrt{f}$ in $L^2$?
Suppose that $\{f_n\}$ is a sequence of measurable functions converging to $f$ in $L^1(\mathbb{R}^n)$. Is it true that $\sqrt{f_n}$ converges to $\sqrt{f}$ in $L^2(\mathbb{R}^n)$?
If this is true ...
1
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1answer
38 views
Cubes covering a set in $\mathbb{R^3}$
Let's say I divided $\mathbb{R^3}$ with 3 mutually orthogonal systems of planes and the distance between two neighboring planes of each system is $\varepsilon$. So basically what I have is countable ...
0
votes
1answer
29 views
fraction of $L^\infty$ functions
Let $\Omega $ be a bounded domain of $R^n$, and let $a, b \in L^\infty(\Omega) $ such that $ : \frac{a}{b} 1_{\{x\in \Omega/ b(x)\ne 0\}} \in L^\infty(\Omega). $
Does the following implication hold ...
1
vote
1answer
22 views
Variation of a signed measure -
I am studying Measure Theory, using the Bartle's book "Elements of Integration and Lebesgue Measure" and I couldn't do the exercise 3.Q:
"If $\mu$ is a charge on $X$, let $\mathcal{v}$ be defined ...
0
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0answers
28 views
A question about Lebesgue measure 3
Let $ L^k $ be the k-dimensional lebesgue measure. Let $ A \subset R^n $ be a Borel set. Suppose we have proved that $ L^1(A \cap l )=0 $ for each line $ l $ parallel to some line passing throught the ...
1
vote
1answer
14 views
How to show $\dim_\mathcal{H} f(F) \leq \dim_\mathcal{H} F$ for any set $F \subset \mathbb{R}$ and $f$ continuously differentiable?
Let $f: \mathbb{R} \to \mathbb{R}$ be differentiable with continuous derivative. I have to show that for all sets $F \subset \mathbb{R}$, the inequality $$\dim_\mathcal{H} f(F) \leq \dim_\mathcal{H} ...
1
vote
1answer
86 views
Lebesgue Integral but not a Riemann integral
Is it possible for a function to be a Lebesgue integral, but not a Riemann integral?
After the comments below I realize my question was not a good one. Thank you.
This is my edited version:
Let $f$ ...
0
votes
2answers
20 views
Doubt in Scheffe's Lemma
While reading up on "Glivenko Cantelli Theorem" from Probability Models by K.B Athreya, the author used 2 lemmae to prove it. One was called Scheffe's lemma, the other Polya's theorem.
Scheffe's ...
