Is the intersection of any subset of $X$ and a $\lambda$-measurable set $\lambda$-measurable?

Question

Suppose $A\subset X$ is $\lambda$-measurable. Is $B\cap A$ $\lambda$-measurable for any $B\subset X$? I have tried on playing with set theories but still cannot come up with a close solution. All I have to do is just show that for any $C\subset X$, $\lambda(C)\geq\lambda(C\cap(A\cap B))+\lambda(C\cap(A\cap B)^c)$. How should I proceed? Thanks!

Answer 1

If that were so, then all subsets would necessarily be $\lambda$-measurable, since $X,$ itself, is $\lambda$-measurable. Simply take $A=X$ to see why.

Thus, it really depends on whether there are any subsets of $X$ that are not $\lambda$-measurable.