# 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 candidate. I'm not certain, but based on answers to related questions, I think this might be the Effros Borel structure that Gerald Edgar has mentioned here and here.

The $\sigma$-algebra $\Sigma$ is an ordered set under the canonical relation given by subset inclusion $\subseteq$, and is therefore naturally equipped with a specialization topology. The closed sets are generated by downward-closed sets, and the closure of a singleton is its down-set:$$\overline{\{A\}} = \{ B \in \Sigma : B \subseteq A \}.$$ Even though this topology is highly non-Hausdorff, it's still pretty nice. For example, it's an Alexandroff space: arbitrary unions of closed sets are closed.

Being a topological space, $\Sigma$ now has a natural measurable structure, namely, the one generated by the Borel $\sigma$-algebra $\Sigma^1 := \mathcal B_{\subseteq}(\Sigma)$.

• Is this space $(\Sigma, \Sigma^1)$ a reasonable one on which to do measure theory and probability?

Whether it is or not, there's some non-trivial structure present. For example, we can iterate this procedure. Set $\Sigma^0 = \Sigma$, and define $\Sigma^n := \mathcal B_{\subseteq}(\Sigma^{n-1}).$ Then each one of these spaces $\Sigma^n(X) := (\Sigma^{n}, \Sigma^{n+1})$ is measurable.

• Is $\Sigma : \mathrm{Meas} \to \mathrm{Meas}$ an endofunctor on the category of measurable spaces?

• Under what conditions does the sequence of measurable spaces $\Sigma^n(X)$ have a limit $\Sigma^{\infty}(X)$?

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Tom, I don't think you mean what you said about Alexandroff spaces; arbitrary intersections of closed sets are always closed, in a topological space. –  Paul McKenney Feb 10 '13 at 4:01
Thanks @Paul McKenney. It was a typo: Alexandroff spaces contain arbitrary unions of closed sets. –  Tom LaGatta Feb 10 '13 at 6:27
One way to approach this would be to ask the same question inside a suitable topos in which "everything is measurable" and such that each object is naturally equipped with the structure of a $\sigma$-algebra. In effect you would be expanding the notion of measure space to accommodate better structure, as such toposes typically contain the "classical" measure spaces.
@Andrej Bauer, that's an interest point of view. Can you expand more on it? Suppose we are considering the category $\operatorname{Meas}$, the topos-category $\operatorname{Set}$ and some other topos $\operatorname{T}$. What does it mean "to ask the same question" in that different topos? –  Tom LaGatta Feb 11 '13 at 1:57