# Interpreting the Loomis--Sikorski theorem

Let $$A$$ be a $$\sigma$$-complete Boolean algebra. By the Loomis--Sikorski theorem, $$A$$ is isomorphic to $$F(X)/M$$, where $$F(X)$$ is the $$\sigma$$-field of Baire subsets of $$X$$ (the Stone space of $$A$$), and $$M\subseteq F(X)$$ is the $$\sigma$$-ideal of meagre sets.

So let $$h:A\to F(X)/M$$ be the isomorphism in question. For every $$a\in A$$, I have a problem interpreting $$h(a)$$. Elements of $$F(X)/M$$ are equivalence classes of Baire sets, so is $$h(a)$$ the representative of an equivalence class of Baire sets? In other words, is $$a\mapsto [h(a)]$$?

• Terry Tao has a quite complete post on this: terrytao.wordpress.com/2009/01/12/…. The isomorphism $h$ is the composition of the isomorphism given by Stone duality (from the Boolean algebra $A$ to the clopen algebra of its Stone space), the inclusion of the clopen sets into the Baire algebra, and the quotient map (mod meagre sets). – Luiz Cordeiro Aug 23 at 15:55
• Why would you think $h(a)$ is a representative of an equivalence class? You said yourself that elements of $F(X)/M$ are equivalence classes, and $h(a)$ is certainly an element of $F(X)/M$. – Eric Wofsey Aug 23 at 19:17

As Terry Tao explains: $$h(a)$$ is the equivalence class (modulo the $$\sigma$$-ideal of meagre and Baire subsets of the Stone space of $$A$$) of the usual Stone-dual image of $$a$$ into the Stone space of $$A$$.
So $$A$$ is in particular a Boolean algebra, hence it has a Stone space $$X=S(A)$$ where each element of $$A$$ corresponds to a unique clopen set $$s(a)$$ of $$X$$ (and all clopen sets of $$X$$ occur as images, and $$a \to s(a)$$ preserves all BA-operations.
In $$X$$ we generate the $$\sigma$$-algebra by the clopen sets, which you call $$F(X)$$. In $$F(X)$$ we consider all meagre sets (that happen to lie in that $$\sigma$$-algebra) and call it $$M$$ and define a quotient $$F(X){/}M$$. We can map each $$a$$ to the equivalence class of $$s(a)$$ (which is clopen and thus lies in $$F(X)$$ obviously) so all sets that differ from $$s(a)$$ by a meagre Baire set; call that class $$h(a)$$. Halmos in the book shows that all classes of members of $$F(X)$$ are reached that way (every class has a unique clopen representative which uniquely corresponds to an element of $$A$$, so it shows $$h$$ is a bijection) and this $$h$$ also preserves all countable operations (and of course all Boolean operations too) and so is an isomorphism of $$\sigma$$-complete algebras.