# Projective family of probability spaces

This is a crosspost of this question from MSE.

I'm confused about the definition of a projective family of probability spaces $(S_t,\mathscr S _t,\mu_t,f_{ts})_{s,t\in T}$. The conditions

• $f_{tt}=1_{S_t}$
• $f_{us}=f_{ts}\circ f_{ut}$ where $s\leq t\leq u$

are clear from the usual definition of a projective system as a functor from $\mathsf T^\text{op}$, the opposite poset category. However, (in addition to measurability) the following condition always appears:

• $\mu_s = (f_{ts})_{\ast}(\mu_t)$, where the RHS is the pushforward measure.

This looks to me like a kind of cone coherence condition, but it does not (as far as I can see) fall out of the categorical formalism of limits if one works in the category with probability spaces as objects and measurable maps as arrows.

Furthermore, the projective limit (if it exists) is required to satisfy the following two conditions:

• Its $\sigma$-algebra is generated by the restrictions of the projections $\pi_t$
• Its probability measure $\mu$ must satisfy $(\pi_t)_\ast (\mu)=\mu _t$

In what category should one work so that all these conditions fall out of the categorical formalism? In other words, in what category does the categorical notion of projective limit coincide exactly with the one found in books.

• How about the category of probability spaces $(\Omega, \mathscr{O}, P)$ where morphisms are measurable maps $(\Omega_0,\mathscr{O}_0,P_0)\to (\Omega_1,\mathscr{O}_1,P_1)$ such that $f_*P_0=P_1$. – Liviu Nicolaescu Mar 31 '15 at 18:57
• @LiviuNicolaescu that feels artificial though. Usually, compatibility conditions come from unraveling the definitions of limit, not from the arrows themselves. – Exterior Mar 31 '15 at 18:59
• Liviu is right. In a category of probability spaces, one wants morphisms that respect the whole probability space structure, not just the $\sigma$-algebra but also the measure. – Andreas Blass Mar 31 '15 at 19:09
• @Exterior I do not understand your comment. The arrows do matter. Think of the ring of $2$-adics $\mathbb{Z}_{(2)}$ vs. the ring of $3$-adics $\mathbb{Z}_{(3)}$. They're defined by identical diagrams, but the meaning of arrows is different. The result is nonisomorphic local rings. – Liviu Nicolaescu Mar 31 '15 at 19:13
• @LiviuNicolaescu I'm sure you are right. Andreas Blass's comment helped me understand that the measure is another piece of structures one should ask the arrows to respect. – Exterior Mar 31 '15 at 19:15

These condition fall out of the categorical formalism, using the category of measurable spaces, provided you view probability measures from a slightly different perspective. Think of a probability measure as an affine, weakly averaging functional which preserves limits $I^X \rightarrow I$, where $I=[0,1]$...see the paper http://arxiv.org/pdf/1406.6030.pdf The category of measurable spaces is a symmetric monoidal closed category which makes all this, from my (biased) perspective, fairly easy to work out the basics. This is indirectly a question about the Giry monad; but the Giry monad is naturally isomorphic to a submonad of the double dualization monad where analysis of this problem is easier. (Giry's paper also attempts to address this projective limits problem also. See http://link.springer.com/chapter/10.1007%2FBFb0092872. It's easier to work in the category of measurable spaces directly (rather than the Kleisi category). Let me just point out that the tensor product of the these spaces in question exist and the projection maps are measurable, so that the pushforward maps look like that in, say Diagram 1, page 9. The projective limit, provided it exist, is then just a probability measure'' on the tensor product space $\otimes_i X_i$, ie., a weakly averaging affine functional $I^{\otimes_i X_i} \stackrel{P}{\longrightarrow} I$ which preserves limits, with the composite $I^{X_j} \stackrel{I^{\pi_j}}{\longrightarrow} I^{\otimes_i X_i} \stackrel{P}{\longrightarrow} I$ being the pushforward along the coordinate projection map $\pi_j$. (These projection maps are measurable on the tensor product space as discussed in the paper.)