# 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 $(\Omega, \Sigma)\rightarrow (\Omega', \Sigma')$ is a function $f$ from $\Omega$ to the set of all probability measures on $\Sigma'$ such that for each $\sigma'$, the function $\omega \mapsto f_\omega(\sigma')$ is measurable. The idea is that $f(\omega)$ is in $\sigma'$ with probability $f_\omega(\sigma')$. So after defining composition: $$(gf)_\omega(\sigma'') = \int_{\Omega'} g_{\omega'}(\sigma'')df_\omega(\omega')$$ (here the notation $\int f(x) d\mu(x)$ indicates that $\mu$ is the measure and $x$ the dummy variable), the condition that composition is associative is equivalent to $$\int_{\Omega'} \left(\int_{\Omega''}h_{\omega''}(\sigma''') dg_{\omega'}(\omega'') \right) df_\omega(\omega')= \int_{\Omega''}h_{\omega''}(\sigma''') d(gf)_\omega(\omega'')$$ Here $f : (\Omega, \Sigma) \rightarrow (\Omega', \Sigma')$, $g : (\Omega', \Sigma') \rightarrow (\Omega'', \Sigma'')$, $h : (\Omega'', \Sigma'') \rightarrow (\Omega''', \Sigma''')$, $\sigma'''\in\Sigma'''$, $\omega \in \Omega$, and the other omegas are dummy variables. Is this known in the literature? I'm not very familiar with measure theory. I strongly suspect that it is true, because all of the other axioms have worked so far (e.g. the composition of two measures really is a measure, identities work out properly). This will probably have a longish, boringish proof involving approximation by simple functions, which I want to avoid going through if someone else has. Indeed, if someone else has proven this, perhaps the category I am considering has already been invented. Does it look familiar to anyone? Perhaps a version of the chain rule, but for integrals?

Edit: The proof of this is actually shorter than I had thought. However, the maps have already been studied; see the answer(s).

Your maps are precisely what is called Markov kernels in probability. These kernels and their composition are discussed in many places. See, for instance, Section 36 from Probability theory by Bauer.

• Knowing they are called "Markov kernels" allows one to find this paper, in which a closely related category is discussed. The f_\omega's are only required to be subprobability measures.
– Luke
Mar 4, 2014 at 12:21
• Of course, one can - it doesn't make much difference (for instance, one can always add an "ideal point" where all the missing probability goes).
– R W
Mar 4, 2014 at 12:56

This is the category of probabilistic mappings, introduced by Lawvere in some never published notes from 1962. This paper by Culbertson and Sturtz has references to most of the subsequent literature and serves as a good primer.