# What is the structure of a space of $\sigma$-algebras?

Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be the evaluation functional, defined by $\delta_x(f) = f(x)$. For any compact $D \subseteq X$, consider the $\sigma$-algebra $$\mathcal F_D = \bigcap_{\epsilon > 0} \sigma \{ \delta_x : x \in D^\epsilon\},$$ where $D^\epsilon$ denotes the $\epsilon$-enlargement of the set $D$ in $X$. Loosely speaking, the $\sigma$-algebra $\mathcal F_D$ encodes the information stored in (an infinitesimal neighborhood of) the set $D$.

The family $\mathbb F_0 = \{\mathcal F_D\}$ admits the obvious partial ordering ($\mathcal F_D \preceq \mathcal F_{D'}$ if $D \subseteq D'$), and moreover is a lattice. It is also a partially-ordered, right-continuous filtration.

Question 1: What is the general structure of the family $\mathbb F_0$?

Question 2: What is the general structure of the space $\mathbb F$ of all $\sigma$-algebras over $\Omega$? Is this space too big to have any nice structure? Is there a more appropriate space to call "the space of $\sigma$-algebras"?

• About question 1: What do you mean by "general structure" ? As you said, it is a filtration, what kind of generalization of this do you expect ? – Pascal Maillard Feb 28 '12 at 11:30
• Motivated by your question, I have finally taken a look into B. Tsirelson, Noise as a Boolean algebra of sigma-fields, arxiv.org/abs/1111.7270 , which has been lying on my desk for weeks :) I think that if there was a good answer to your question, it would probably appear there. But as far as I understand, the only "general structure" of $\mathbb F$ (or $\mathbb F_0$) that appears there is that of a distributed lattice, with $\mathbb F_0$ being almost a Boolean algebra (it would be a Boolean algebra if you took the sets $D$ instead of $D^\epsilon$ in the definitions. – Pascal Maillard Feb 28 '12 at 11:50
• Thanks for the nice reference, Pascal. I'm always a fan of Tsirelson's work. So a space of $\sigma$-algebras is fundamentally a distributive lattice, but one needs to impose a probabilistic structure on top of that to have a meaningful continuous-type noise. Here's another Tsirelson paper on the topic: projecteuclid.org/… – Tom LaGatta Mar 4 '12 at 21:54
• @Tom: Thank you too for the nice reference. By the way, if you put @Pascal at the start of your comment, I get notified by MO (I just learned this reading the comments of this question: mathoverflow.net/questions/90218/… ) – Pascal Maillard Mar 6 '12 at 13:12
• @Pascal ^^ (in case the end of the sentence doesn't work) – Tom LaGatta Mar 8 '12 at 7:05