Questions tagged [measure-theory]
Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.
19 questions from the last 7 days
-2
votes
1
answer
198
views
+200
Pigeonhole principle based measure
Define the upper pigeonhole density of some $X\subset \mathbb R^d$ as
$\overline{pd}(X)=\limsup_{n\in \mathbb N} \inf_{S\subset\mathbb R^d, |S|=n} \sup_{x\in R^d} \frac{|S\cap (X+x)|}{|S|}.$
Can I ...
- 882
0
votes
1
answer
95
views
What is wrong with my calculation? (Inclusion-Exclusion-Principle)
Consider the alphabet $\mathcal A:=\{a,b\}$ and a uniformly drawn word with three letters $X:\Omega\to\mathcal A^3$ (note that from this we already have that the marginal distributions must be ...
- 668
3
votes
1
answer
104
views
Radon measures in the book by Adams & Hedberg
I am currently trying to figure out what is a Radon measure in the book "Function spaces and potential theory" by Adams and Hedberg. Let me paraphrase the definition (Section 1.1.3 on page 2)...
- 33.9k
3
votes
1
answer
106
views
Find a.e. pointwisely bounded subsequence in a bounded sequence in $L^p$
Let $f_i$ be a bounded sequence in $L^p([0,1])$ where $p\in[1,\infty)$, i.e. $\sup_i\|f_i\|_{L^p}<+\infty$. Does there always exist a subsequence $f_{i_k}$ such that
$$\left|\left\{x\in[0,1]:\...
- 12k
3
votes
1
answer
68
views
Approximate set of finite Lebesgue measure by finite union of open intervals
This problem is from Bartle.
Let $A \subset \mathbb{R}$ and $\lambda$ the Lebesgue measure. If $\lambda(A)$ is finite, then, for every $\varepsilon > 0$, there is a set $G$ which is the union of a ...
- 1,265
0
votes
1
answer
57
views
Measurability of function minimizing an integral
For my question I am trying to simplify the setting of concern as much as possible.
Therefore, let $(\Omega, \mathcal{A}, P)$ be a probability space and $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}...
- 766
1
vote
1
answer
70
views
Equivalence of two definitions of finite complex measures
I am trying to understand Remark 4.29 in Pavlov's paper. He defines measures as follows.
Definition 4.28. A (complex infinite) measure on an enhanced measurable space $(X,M,N)$ is a map $\mu:M' →\...
- 119
3
votes
0
answers
78
views
Borel-$\sigma$ algebra on a manifold
Let M be an n-dimensional manifold, and let $(\pi_\alpha: U_\alpha \to V_\alpha)$ be an atlas of coordinate charts for M, where $U_\alpha$ is an open cover of M and $V_\alpha$ are open subsets of ${\...
- 1,869
0
votes
1
answer
70
views
The product of measurable functions $f(x)$ and $g(y)$ is measurable on $X \times Y$
This problem is from Bartle.
Let $(X, \mathscr{A})$ and $(Y, \mathscr{B})$ be measurable spaces. If $f$ is an $\mathscr{A}$-measurable function and $g$ is a $\mathscr{B}$-measurable function, then $h ...
- 1,265
3
votes
1
answer
52
views
Understanding the Borel-Stieltjes measure
I'm reading Bartle's Elements of integration and I'm confused about his definition of the Borel-Stieltjes measure. Paraphrasing the context: for an increasing function $g \colon \mathbb{R} \to \mathbb{...
- 1,265
2
votes
0
answers
65
views
Generalizing integration of $[0,+\infty]$ valued functions
Several of the most basic results of Lebesgue integration are stated for nonnegative (Lebesgue measurable) maps $X \to [0,+\infty]$ and may involve the order structure $\leq$ of $[0,+\infty]$:
Fatou'...
- 21.3k
1
vote
0
answers
39
views
Preimage of generator is in sigma algebra
Let $A$ be a countable set of indices and for every $a\in A$ let $M_a$ be a $\sigma$-algebra on a non empty set $X_a$, generated by a family of subsets $\epsilon_a$, that is $M_a=\sigma(\epsilon_a)$. ...
- 65
2
votes
0
answers
39
views
Area estimate of the intersection of ball and sphere
The following estimate arises in the proof of Tomas-Stein restriction theorem.
$$
\sigma_{\mathbb{S}^{d-1}} (B(x,r)\cap \mathbb{S}^{d-1}) \leq C r^{d-1}
$$
The estimate is very intuitive, and I have a ...
- 666
0
votes
1
answer
38
views
Right continuity of $f$ and bounded variation on $[a,b]$ imples right continuity of $V(a,t)$
I'm studying from Bobrowski Functional analysis for probability and stochastic processes (not a university course). I got stuck on one of the exercises, exercise 1.3.4.
Let $f:[a,b]\to \mathbb{R}$ ([a,...
- 11
1
vote
1
answer
51
views
Dominated statistical models
This is a simple question from measure theory.
Fix a measurable space $(E,\mathcal{E})$ and a family $(P_i)_{i\in I}$ of probability measures on $(E,\mathcal E)$ ($I$ is any non-empty set). Let $n\geq ...
- 451