I'm dealing with the random forcing $\mathbb{P}=\mathbb{M}\mathbb{B}(X,\mu)$ defined in Kunen's 2013 book. Recall that the conditions of that forcing are the equivalence classes of mesurable sets modulo null sets $[S]$ endowed with the order $[S]\leq [T]$ iff $S\setminus T\in \text{null}$.
In this regard I'm interested to see that kind of forcing has the ccc property. To see this I'm trying to prove that the sets $\{q\in A: \mu(q)\geq 1/n\}$ have cardinality at most $n$. My idea was related with take $n+1$ unions of members of that sets and get a contradiction. The problem is that I don't know if $p$ and $q$ are incomparables implies that their corresponding $S$ and $T$ are disjoint so I'm not willing to make work my argument. Have anybody any idea? Is this a adequate strategy? If not, could you give me any proof or advice?
Thanks in advance.
First, there's a slight typo in your definition of the random forcing: presumably you want positive measure measurable sets to be conditions. Also, I assume you want the whole space to have measure $1$ (rather than infinity).
As to the ccc property, you're basically on the right track. If $[S], [T]$ are incompatible, then $S$ and $T$ may not be disjoint, but they will be almost disjoint in the measure sense: $S\cap T$ will be null (why?). Note that "almost disjoint" usually means "disjoint except for a finite set".
So suppose I have an uncountable antichain.
I can find an uncountable sub-antichain consisting of elements of measure $>r$, for some positive real $r$ (why?).
WLOG, $r={1\over n}$ for some $n$; so pick $n+1$-many elements $S_1, . . . , S_{n+1}$ of this sub-antichain.
Now the measure of the union $\bigcup_{i\in \{1, 2, . . . , n+1\}} S_i$ is at least the sum of the measures of the $S_i$s, minus the measures of their pairwise intersections (why?).
But this is ${n+1}\over n$, since the pairwise intersections are null; so we have a contradiction.