Skip to main content

All Questions

Filter by
Sorted by
Tagged with
3 votes
2 answers
199 views

An uncountable measurable subset of $\Bbb R$ containing no nonempty perfect set

$\newcommand\R{\Bbb R}$Assuming the axiom of choice, is there an uncountable Lebesgue-measurable subset $S$ of $\R$ that contains no nonempty perfect set? Of course, such a set $S$, if it exists, ...
Iosif Pinelis's user avatar
11 votes
1 answer
500 views

Uncountable families of measurable sets with pairwise positive intersections

Let $(X,\mathcal{B},\mu)$ be an arbitrary finitely additive probability measure space, let $a>0$ and let $(A_i)_{i\in I}$ be an uncountable family of subsets with measure $\geq a$. Is there an ...
Saúl RM's user avatar
  • 10.6k
4 votes
1 answer
204 views

How probability-rich is the $\sigma$-algebra generated by a sequence of sets? (Sierpiński's theorem on non-atomic measures without using the AoC.)

$\newcommand\F{\mathcal F}\newcommand\si{\sigma}\newcommand\Om{\Omega}\newcommand\ep{\varepsilon}$Let $p\in(0,1)$ and let $(\Om,\F,P)$ be a probability space. Let $(A_n)$ be a sequence in $\F$ such ...
Iosif Pinelis's user avatar
8 votes
2 answers
960 views

Is there a measure theory for proper classes?

This question is naive, but I didn't get an answer at MSE: Is it straightforward to extend measure theory to proper classes? Of course when one tries to define measures on "large sets" ...
aduh's user avatar
  • 869
4 votes
1 answer
259 views

Reference request: large generalized probability measures

I'm interested in references relevant to the following: what is the right generalization, if there is one, of a probability measure that takes on values in an structure of more than continuum size? I'...
Beau Madison Mount's user avatar
2 votes
0 answers
320 views

Showing that the procedure to generate an algebra does not generate a $\sigma$-algebra

A while ago I asked this question on mathstackexchange: Let $\mathscr{C}\subset \mathscr{P}(\Omega)$ be a class of subsets of a nonempty set $\Omega$ containing $\Omega$ and $\varnothing$. Define $\...
Eduardo's user avatar
  • 757
8 votes
4 answers
775 views

Self-contained formalization of random variables?

I have not been able to find any formalization of random variables that supports construction of new random variables dependent on previously constructed ones. In what I have found, a random variable $...
user21820's user avatar
  • 2,912
12 votes
1 answer
316 views

A reference to a theorem on the equivalence of ideals of measure zero in the Cantor cube

I am looking for a reference of the following (true) fact: Theorem. For any two continuous strictly positive Borel probability measures $\mu,\lambda$ on the Cantor cube $2^\omega$ there exists a ...
Taras Banakh's user avatar
  • 41.8k
19 votes
6 answers
3k views

Sierpinski's construction of a non-measurable set

In the early 20th century there was a lot of fuss over the axiom of choice implying that there are Lebesgue non-measurable sets of reals. In his book about The Axiom of Choice, Gregory Moore points to ...
Asaf Karagila's user avatar
  • 39.7k