2
votes
1answer
196 views

Are measures of a measurable cardinal measurable? (Edited and Updated Version)

Update: Regarding to Prof. Hamkins's guidance I restricted the questions to the "normal" measures to avoid trivial answers. Definition: Let $\kappa$ be a measurable cardinal. Define: ...
23
votes
3answers
782 views

Does every set of reals contain a measure-zero set of the same cardinality? Does it contain a meager set of the same cardinality?

This question arises from an issue in my post on Ashutosh's excellent question on Restrictions of the null/meager ideal. Question 1. Does every set of reals contain a measure-zero subset of the same ...
16
votes
1answer
455 views

How strong is “all sets are Lebesgue Measurable” in weaker contexts than ZF?

Famously, Solovay showed that, if $\textrm{ZFC}$ plus $\textrm{IC}$ (the existence of an inaccessible cardinal) is consistent, then so is $\textrm{ZF}$ plus $\textrm{DC}$ (dependent choice) plus ...
9
votes
1answer
446 views

Is there a probability theory developed in intuitionistic logic?

Since Boole it is known that probability theory is closely related to logic. According to the axioms of Kolmogorov, probability theory is formulated with a (normed) probability measure ...
3
votes
2answers
321 views

Cohen algebra (generalization)

Let Bor($X$) = class of all borel subsets of $X$. Cohen algebra is defined as Bor(X) modulo the ideal of meager sets. The Cohen algebra has a combinatorial : it is the unique atomless complete ...
4
votes
2answers
612 views

Are all models of ZF + DC + “All set of reals are lebesgue measurable” also models of CH? [duplicate]

Possible Duplicate: Lebesgue Measurability and Weak CH I have studied a little set theory and I found that Solovay constructed a model of ZF+DC+"All set of reals are Lebesgue measurable" and ...
11
votes
2answers
635 views

Measuring big stuff

Often during informal discussion with colleagues, the following pattern emerges when we are stuck trying to prove a theorem about $x \in X$. A: "let's assume this hypothesis $H$ on $x$" B: "most ...
20
votes
4answers
2k views

Nonstandard analysis in probability theory

I am quite new at nonstandard analysis, and recently I became aware of its use in probability theory mainly through the following two books: Nelson (1987). Radically Elementary Probability Theory ...
7
votes
1answer
542 views

Intuition behind the diagonal intersection

Suppose that for all $\alpha<\kappa$ we have that $A_\alpha\subseteq\kappa$. We define the diagonal intersection to be $$\bigtriangleup_{\alpha<\kappa}A_\alpha = \left\lbrace\xi<\kappa\ ...
18
votes
4answers
1k views

Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?

One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...
6
votes
1answer
387 views

Probability that a Turing machine will nontrivially reduce a real

For a fixed Turing machine $\Phi_e$, what is the probability that it will reduce a given real to some less complex, yet still non-computable real? More precisely: It is known that the set of reals ...
9
votes
1answer
449 views

Can we change the Lebesgue measure by forcing?

Suppose $M$ is a model of ZFC, and $\mu^M$ is the Lebesgue measure on $\mathbb R^M$ such that $\mu^M(\mathbb R^M)=1$. It is known that if $r$ is a Cohen real over $M$ and $N=M[r]$ then $\mu^N(\mathbb ...
0
votes
1answer
926 views

basic measure theory question - measure on the natural numbers [closed]

I am looking for a succinct way to describe a subset of the natural numbers which has ``measure zero" in the following sense: Let X_1 \subset X_2 \subset .... be any strictly nesting sequence of ...
18
votes
1answer
645 views

A collection of intervals that can cover any measure zero set

This is a follow-up to this question (in fact, this is what originally motivated me to ask that one.) Let's say that a sequence $\{s_i\}$ of positive reals covers a set $X\subset\mathbb R$ if there ...
17
votes
2answers
1k views

A set that can be covered by arbitrarily small intervals

Let $X$ be a subset of the real line and $S=\{s_i\}$ an infinite sequence of positive numbers. Let me say that $X$ is $S$-small if there is a collection $\{I_i\}$ of intervals such that the length of ...
18
votes
4answers
1k views

Is every probability space a factor space of the Haar Measure on some group?

Let P be an arbitrary probability space. I would like to find a compact topological group $G$ so that the Haar probability measure on $G$ admits a measurable map to the probability space $P$. By a ...
6
votes
1answer
839 views

Correspondence between functions on a set and “states” on its power set

Let $L$ be the poset (ordered by set inclusion) that is the power set of some set $X$. A state is a function $s:L \rightarrow [0,1]$ satisfying i) for {$p_1,p_2,...$}, $p_i \in L$ a pairwise ...