Show that every measurable set $A$ can be written $A=B \cup C$ where $B=\cap_{i=1}^\infty O_i$ ($O_i$ are open) and $C$ is measure $0$.
How should one approach this and can someone refer me to a good text that addresses expressing a measurable set in terms of a $C$ with measure $0$?
I didn't read your problem carefully the first time. This is false: Take for example a meager set A of reals whose complement is null set.
It seems to me that you wanted to get a $G_{\delta}$ set containing $X$ such that the residual is null. This is a consequence of regularity of Lebesgue measure. A proof will depend of what a measurable set of reals is to you. One standard definition is (see Oxtoby for example): X is measurable if for every $\epsilon > 0$ there exist closed and open sets $F_{\epsilon}, U_{\epsilon}$ resp. such that $F_{\epsilon} \subseteq X \subseteq U_{\epsilon}$ and $\mu(U_{\epsilon} \backslash F_{\epsilon}) < \epsilon$. In this case the proof is trivial, by taking the intersection of $U_{\epsilon}$ as $\epsilon$ runs over all positive rationals gives you the desired $G_{\delta}$ cover. If this isn't your definition then you should state your definition (maybe it is Caratheodory outer measurable) and I will supply the necessary argument.