Borel algebra mod meagre sets vs. Borel algebra mod sets of measure zero

Question

Are there any circumstances in which the Borel algebra modulo meagre sets is isomorphic to the Borel algebra modulo sets of measure zero?

Answer 1

Measure and Category by Oxtoby, in chapter 22,considers this problem. First he sketches how you cannot find such isomorphisms for many metric spaces, and then defines a Category measure as topological spaces with a measure such that the sets of measure $0$ and those of first category (meagre sets) are the same.

He then shows that one can sometimes define a topology on a measurable space to make it such a measure space. An important example is the density topology on $\mathbb{R}$. I also know of Kunen's compact $L$-space under CH (in Fund. Math.) that obeys this.