Linked Questions
31 questions linked to/from Is there an introduction to probability theory from a structuralist/categorical perspective?
8
votes
1
answer
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What's the point of a point-free locale?
In [1, example C.1.2.8], a locale $Y$ (dense in another locale
$X$) without any point is given. I fail to understand the point
of such point-less locale - Why can't we identify those as the
trivial ...
- 5,230
4
votes
3
answers
800
views
Quick derivation of classical probability theory from von Neumann algebraic framework
Watching (the begining of) a lecture on free probability theory by Dimitri Shlyakhtenko https://www.youtube.com/watch?v=F8Urtr39jM0, I'm led to consider the following question
Question. How can one ...
- 6,853
7
votes
2
answers
1k
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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 $(X,\mathcal{B},\nu)$ ...
- 193
6
votes
2
answers
1k
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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 ...
- 1,344
21
votes
1
answer
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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 ...
- 37.8k
20
votes
1
answer
686
views
A nice subcategory of the category of measurable spaces
Is there some notion of "nice" measurable spaces and "nice" maps between them which satisfies the following properties?
The real line equipped with the Lebesgue $\sigma$-algebra is nice.
Any ...
- 301
5
votes
1
answer
358
views
Is there a meaningful interpretation of an $L^i$-space?
Do complex-normed spaces exist? Is there an extension of $p$-norms to $p\in\Bbb C\setminus\Bbb R$?
A while ago I thought of extending $L^p$-spaces to the complex-normed setting. After some discussions,...
- 1,459
7
votes
1
answer
430
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Is the Pierce spectrum useful elsewhere in Mathematics?
In Borceaux and Janelidze's Galois Theories, a construction of the Pierce spectrum is given. It is the poset of ideals in a Boolean ring. It's construction is reminiscent of the Zariski spectrum in ...
- 2,369
4
votes
1
answer
609
views
Notation: Categories of measur(abl)e spaces
Is there a common notation in the literature for
the category of measurable spaces and measurable maps?
the category of measure spaces and measure-preserving maps?
The nlab suggests $\mathsf{Measble}...
- 63.1k
11
votes
1
answer
483
views
Terminology for this notion of "$\sigma$-algebra" in a topos
Let $\mathcal{E}$ be a Grothendieck topos. I want to define a sort of "$\sigma$-algebra" for it, and I'm asking about what it should be—or already is—called. I know from nlab that Cheng spaces are an ...
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4
votes
1
answer
309
views
Categorified probability and statistics?
To put it simply, what if the sample space underlying our probability space is a category instead of a mere set. Has a theory or probability and statistics been developed for such a situations in ...
- 803
2
votes
0
answers
468
views
A Grothendieck style reference for probability theory
This is a very vague question.
I am looking for an "EGA" type of reference for probability theory. This means (among other things) that I'm looking for a text which develops the theory "abstractly".
- 29
5
votes
1
answer
217
views
Dense subcategory of measurable spaces
Recall the notion of a dense subcategory $\mathcal{D}$ of a category $\mathcal{C}$. It means that the restricted Yoneda functor $\mathcal{C} \to \mathrm{Hom}(\mathcal{D}^{op},\mathbf{Set})$, $A \...
- 63.1k
4
votes
1
answer
226
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Definition of Radon measure on Takesaki's first volume
Consider the following theorem from Takesaki's first volume "Theory of operator algebras":
In $(i)$, it is claimed that $L^\infty(\Gamma,\mu)$ is an abelian von Neumann algebra. How does ...
- 175
3
votes
1
answer
145
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Reference for "the algebra of multiplication by all measurable bounded functions acts in Hilbert space in a unique manner"
I read a paper of Alain Connes on "Duality between shapes and spectra" and in page 4, he says
Due
to a theorem of von Neumann the algebra of
multiplication by all measurable bounded ...
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