# Tagged Questions

**2**

votes

**1**answer

197 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:
...

**22**

votes

**3**answers

788 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

**1**answer

462 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 ...

**10**

votes

**1**answer

457 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

**2**answers

323 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

**2**answers

616 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

**2**answers

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**

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**4**answers

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

**1**answer

549 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

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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

**1**answer

388 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

**1**answer

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

**1**answer

941 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**

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**1**answer

647 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**

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**2**answers

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 ...

**19**

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**4**answers

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

**1**answer

843 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 ...