Cardinality of an algebra

Question

Suppose that $B$ is the Boolean algebra of all Lebesgue measurable sets in $I=[0,1]$ modulo Null sets.

I am asking

(1) What will be the cardinality of $B$. Does it have to be $|B|=\mathfrak c$.

(2) Is there any $b\in B$ which is not Borel set.

Answer 1

The cardinality is $\mathfrak c$: First of all, it is at least $\mathfrak c$, as the sets $(0,x)$ are all inequivalent for different values of $x\in[0,1]$. Second, Lebesgue measure is regular, so any measurable set contains a $\sigma$-compact subset of the same measure, and is contained in a $G_\delta$ superset of the same measure. This shows that each equivalence class has a Borel representative. But there are only $\mathfrak c$ Borel sets.

Answer 2

Yes given $X$ Lebesgue measurable take the intersection of a countable sequence of open sets $O_n \supseteq X$ such that $m(O_n) \leq m(X) + \frac{1}{n}$. Then $m(\bigcap O_n-X)=0$ and $X\equiv \bigcap O_n$ in $B$. This means that any equivalence class in $B$ contains a $G_{\delta}$ set.