Structure of the $L_1$ space of measurable subsets of $[0,1]$

Question

Let $\mathcal A$ be a Borel $\sigma$-algebra on $[0, 1]$, and let's introduce a metric on it by $$ d(A, B) = \lambda(A\mathbin\Delta B) \qquad \forall A,B\in \mathcal A $$ where $\lambda$ is the Lebesgue measure and $\Delta$ is the symmetric difference. If we take a quotient space modulo $A\sim B \iff d(A,B) = 0$, do we get some other well-known metric space? Of course, it is a subspace of $L_1([0,1])$ , but I hoped for a more concrete description. I also think, that space may be somewhat smaller. For example, if we consider a discrete measure with support of $n$ points, then we get $\Bbb R^n$ for functions and $2^n$ for measures.

Answer 1

I think that almost everything you might want to know about this space ($\mathrm{MALG}_\lambda$, called the Lebesgue measure algebra) is in exercises (17.42)-(17.46) in Kechris' book.

This metric $d$ turns $\mathrm{MALG}_\lambda$ into a Polish space, and the Boolean operations ($\cup$, $\cap$, and relative complement in $[0,1]$) are well defined and continuous.

Moreover, $\mathrm{MALG}_\lambda$ is the unique, up to iso, measure algebra which, as a $\sigma$-Boolean algebra

1. is generated by a countable set and
2. has no atoms.

As an aside, an analogous algebra can be defined in terms of category, but there is no Polish topology making the Boolean operations continuous for this one.