# Tagged Questions

**2**

votes

**2**answers

123 views

### Integral over integrals with different measures

I was trying to construct a category of measurable spaces and 'nondeterministic functions' (not the usual category of measurable spaces); specifically a map $(\Omega, \Sigma)\rightarrow (\Omega', ...

**4**

votes

**2**answers

173 views

### Obtaining conditional probabilities as pushforwards of [0,1]

It is standard that every Borel probability measure on a polish space $X$ can be obtained as pushforward of the uniform measure $\lambda$ on $[0,1]$ along an almost-everywhere-defined Borel-measurable ...

**4**

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**0**answers

130 views

### Beck-Chevalley for measures?

A measurable set is a pair $(X,\Sigma)$ where $X$ is a set and $\Sigma$ is a $\sigma$-algebra on $X$. The elements $U\in\Sigma$ will be considered as subsets $U\subseteq X$. A morphism of measurable ...

**4**

votes

**1**answer

90 views

### A terminal coalgebra of a certain functor on Mes

Let $\mathfrak C = \mathsf{Mes}$ be the category of meausurable spaces and measurable maps. For any object $X\in \mathfrak C_0$ we assign a measurable space $\mathcal P(X)$ whose elements $\mu$ are ...

**9**

votes

**1**answer

289 views

### Is there a natural measurable structure on the $\sigma$-algebra of a measurable space?

Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space?
Here is one natural ...

**11**

votes

**2**answers

525 views

### Can one view the Independent Product in Probability categorially?

One can construct a category of probability spaces, but this category has no products. Now probability theory relies strongly on the ability to build independent products, the product measure. In a ...

**7**

votes

**2**answers

350 views

### Fixed objects of the M endofunctor on category Meas

Consider the category $\operatorname{Meas}$ of measurable spaces: its objects are sets equipped with $\sigma$-algebras, and its morphisms are measurable functions between spaces.
As Gerald Edgar ...

**6**

votes

**1**answer

287 views

### Groupoids and hypergroups

There are two generalizations of usual groups: groupoids, where the multiplication operation becomes "partial", and hypergroups, for which the result of multiplying two elements is a probability ...

**5**

votes

**0**answers

181 views

### Diffusion processes in wide generality

It is common knowledge among schoolchildren that one may define jump diffusion processes in wide generality.
Hard question: What are the most general structures on which one may define something ...

**5**

votes

**1**answer

363 views

### Which categorical (coproduct-like) operation captures integration of measures?

Suppose we have a measure space $(X,a)$, a measurable space $Y$ and for every $x\in X$ we have a measure $b_x$ on $Y$. Suppose that $(Z,c)$ is a measure space such that as a measurable space ...

**18**

votes

**2**answers

3k views

### Is there a category structure one can place on measure spaces so that category-theoretic products exist?

The usual category of measure spaces consists of objects $(X, \mathcal{B}_X, \mu_X)$, where $X$ is a space, $\mathcal{B}_X$ is a $\sigma$-algebra on $X$, and $\mu_X$ is a measure on $X$, and measure ...

**12**

votes

**4**answers

1k views

### measure spaces as presheaves?

I recently had the idea that maybe measure spaces could be viewed as sheaves, since they attach things, specifically real numbers, to sets...
But at least as far as I can tell, it doesn't quite work ...