Linked Questions

228 votes
13 answers

Is there an introduction to probability theory from a structuralist/categorical perspective?

The title really is the question, but allow me to explain. I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of ...
Pete L. Clark's user avatar
49 votes
28 answers

Problems where we can't make a canonical choice, solved by looking at all choices at once

It's a common theme in mathematics that, if there's no canonical choice (of basis, for example), then we shouldn't make a choice at all. This helps us focus on the heart of the matter without giving ...
172 votes
7 answers

Why do probabilists take random variables to be Borel (and not Lebesgue) measurable?

I've been studying a bit of probability theory lately and noticed that there seems to be a universal agreement that random variables should be defined as Borel measurable functions on the probability ...
Mark's user avatar
  • 4,744
60 votes
7 answers

Is there a measure zero set which isn't meagre?

A subset of ℝ is meagre if it is a countable union of nowhere dense subsets (a set is nowhere dense if every open interval contains an open subinterval that misses the set). Any countable set ...
Anton Geraschenko's user avatar
48 votes
7 answers

What's the use of a complete measure?

A complete measure space is one in which any subset of a measure-zero set is measurable. For what reasons would I want a complete measure space? The only reason I can think of is in the context of ...
Tom E's user avatar
  • 481
47 votes
4 answers

When is $L^2(X)$ separable?

I have never studied any measure theory, so apologise in advance, if my question is easy: Let $X$ be a measure space. How can I decide whether $L^2(X)$ is separable? In reality, I am interested in ...
Bugs Bunny's user avatar
37 votes
5 answers

Reference for the Gelfand duality theorem for commutative von Neumann algebras

The Gelfand duality theorem for commutative von Neumann algebras states that the following three categories are equivalent: (1) The opposite category of the category of commutative von Neumann ...
Dmitri Pavlov's user avatar
26 votes
3 answers

Why is free probability a generalization of probability theory?

Note: This question was already asked on Math.SE nearly a week and a half ago but did not receive any responses. To the best of my knowledge, free probability is an active topic of research, so I hope ...
Chill2Macht's user avatar
  • 2,567
15 votes
3 answers

Integration of differential forms using measure theory?

Setup: Let $(M,g)$ be a (possibly non-compact) Riemannian manifold with volume density $d_gV$. Then one may think of $(M,g)$ as a measure space $(\Omega,\mathcal{A},\mu)$, where $\Omega:=M$, $\mathcal{...
Meneldur's user avatar
  • 398
18 votes
4 answers

Monoidal structures on von Neumann algebras

My question is based on the following vague belief, shared by many people: It should be possible to use von Neumann algebras in order to define the cohomology theory TMF (topological modular forms) in ...
André Henriques's user avatar
7 votes
2 answers

Conditional Expectation for $\sigma$-finite measures

Someone knows of some definition or reference of how to define conditional expectation for a measure space with $\sigma$-finite measure. I think it should be as follows: Let $(X,\mathcal{B},\nu)$ ...
Rusbert's user avatar
  • 173
3 votes
3 answers

Quick derivation of classical probability theory from von Neumann algebraic framework

Watching (the begining of) a lecture on free probability theory by Dimitri Shlyakhtenko, I'm led to consider the following question Question. How can one ...
dohmatob's user avatar
  • 6,546
5 votes
2 answers

Commutative von Neumann algebras and localizable measure spaces

This is not my subject so I apologize if my question is too obvious or understood from other pages. I read some pages such as Reference for the Gelfand-Neumark theorem for commutative von Neumann ...
Ilan Barnea's user avatar
  • 1,324
21 votes
1 answer

Which complete Boolean algebras arise as the algebras of projections of commutative von Neumann algebras?

Projections in an arbitrary commutative von Neumann algebra form a complete Boolean algebra. Moreover, a morphism of commutative von Neumann algebras induces a continuous morphism of the corresponding ...
Dmitri Pavlov's user avatar
7 votes
1 answer

Subfactor theory and Hilbert von Neumann Algebras

There seem to be intimate connections between the different definitions of von Neumann module. The two that I'm aware of are Hilbert von Neumann modules and correspondences (in the sense of Connes). I ...
Ollie's user avatar
  • 1,381

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