Suppose that $\lambda$ be a measure on the interval $I=[0,1]$, and Let $\mathcal{N}$ be the family of null sets. It is known that "measure algebra $\mathcal{B}$ " is the algebra of all measurable subsets of $I$ modulo $\mathcal{N}$, the elements in $\mathcal{B}$ are equivalence classes [.] determined by $\sim$ as:
$A\sim B $ iff $A\Delta B\in \mathcal{N}$ for every measurable sets $A,B$ in $I$
The class $[B]=\{A:A \text{ is measurable }, A\sim B\}$.
My question is: Is it true that these class have the property that every two classes either have empty intersection or they are equal. If yes, it means that we do not have a subclass as $[A]\subseteq[B]$.
Please someone explain this further for me.
Yes. To see this, you only need to show ~ is an equivalence relation. This boils down to showing that if $U$ and $V$ have measure zero, then so does $U \Delta V$.