Questions about abstract measure and integral theory. Also concerns such properties as measurability of maps and sets.
-1
votes
0answers
17 views
Is the countable sum of a set of measurable functions also measurable? [migrated]
It is rather straightforward to show that the sum of two measurable functions is also measurable. Therefore we can extend the logic to say that $\sum\limits_{i=1}^n f_i$ is measurable providing $f_i$ ...
0
votes
0answers
27 views
Every Borel set is the union of an increasing sequence of Bounded Borel sets? [migrated]
I am currently working with the book by Halmos, and i can't quite get past this one.
It states that:
"Every Borel set can be written as an increasing sequence of Bounded Borel sets"
In this case $X$ ...
1
vote
0answers
104 views
When does a proper Zariski closed set have measure zero with respect to a conditional measure?
Assume we have a probability measure $\mu$ over $\mathbb{R}^d$ that is absolutely continuous with respect to Lebesgue measure.
Given $m$ polynomials $p_1,\ldots,p_{m}\in \mathbb{R}[x_1,\ldots,x_d]$ ...
2
votes
1answer
95 views
Is there a survey of recent work relating to the Hausdorff dimension of sets defined through some restriction of digits?
I am familiar with the work of Helmut Cajar, but his book is thirty years old and it's clear that there has been substantial progress since then. I have been spending a lot of time looking through ...
2
votes
0answers
36 views
Possible Hausdorff dimension of intersection of Besicovitch-Eggleston like sets
Let $b \geq 2$ be an integer and suppose that $v=(p_0,\cdots,p_{b-1})$ be a probability vector. Let $S_{b,v}$ be the set of real numbers whose $b$-ary expansion has the digit $k$ with relative ...
3
votes
1answer
135 views
Extending Tarski's Theorem on invariant measures
Tarski's Theorem says that if $G$ acts on $X$ and $E$ is a non-$G$-paradoxical subset of $X$, then there is a finitely additive $G$-invariant measure $\mu:2^X\to[0,\infty]$ with $\mu(E)=1$.
I am ...
2
votes
1answer
233 views
A reference for this possibly well-known fact concerning the Kakeya conjecture?
I believe I have read or heard somewhere that the Kakeya conjecture would follow from appropriate lower bounds for the minimal size of a subset of $\{ 1 , \cdots , N\}$ which contains a translate of ...
1
vote
2answers
177 views
Reference request: learn measure theory for PDEs
I am requesting some references to learn appropriate measure theory for PDEs. Specifically, I would like to learn all the measure theory necessary to understand well-posedness of PDEs with measure ...
0
votes
0answers
111 views
Is there a translation invariant measure on an infinite dimensional space 'without points'?
This is just a reference request. I thought I'd come across a paper demonstrating that there is a translation invariant measure on an infinite-dimensional space without 'points' whilst browsing the ...
0
votes
0answers
24 views
Density invariant of Quotient
Let $G$ be a locally compact group with left Haar measure $\mu$. Furthermore, $\Gamma_1, \Gamma_2 \subset G$ are such that $\mu$ induces finite Haar measures $\mu_1$ and $\mu_2$ on $G /\Gamma_1$ and ...
0
votes
1answer
60 views
identically distributed random variables and measure-preserving transformations
Let $X$ and $Y$ be identically distributed bounded random variables defined on a probability space $(\Omega,\mathcal{F}, \mathbb{P})$. I want to know if there always exists an invertible ...
4
votes
1answer
107 views
Winning sets of full measure (Schmidt's game)
A quick reminder of the definition of Schmidt's game:
Let ${X}$ be a metric space and ${S\subset X}$ be a subset. Let
${0<\alpha,\beta<1}$ be constants. Bob chooses any open ball
...
1
vote
1answer
79 views
Density of certain functions in $C_c^\infty(0,T;V)$ in the space $W(0,T) \approx H^1(0,T;V)$?
EDIT: I need to think more about the question I want to ask given comments in the answer below. Please close the thread if required. I leave it undeleted because answer is useful.
Let $V \subset H ...
9
votes
2answers
356 views
Why do we want maps to be measurable (in countably-additive setting)
When I have to explain things that I am doing to people who did not do (or even did not learn) measure-theoretical probability, I think of getting a question in the title, and I am not sure I have ...
5
votes
0answers
127 views
Existence of an universally measurable pullback
Edited: previous version of the question was less general but also less readable.
Let $X,Y$ and $Z$ be standard Borel spaces, that is topological spaces homeomorphic to Borel subsets of complete ...