Show that every lebesgue measurable subset of R is the union of a Borel subset of R and a set of of outer measure zero.
Let E be a lebesgue measurable subset of R
For n = 1,2,3,... we can find a set of open intervals {$\displaystyle I^n_k$} suvh that $\displaystyle \sum_{k=1} {m(I^n_k)} \leq m(E) + 1/n $
Let $\displaystyle G_n = \bigcup_{k=1}{I^n_k} $. Then $\displaystyle G_n $ is borel since it is the union of open sets.
Let $\displaystyle G = \cap{G_n} $, then G is also Borel.
Finally, I claim m(G\E) = 0, it suffices to show m(E) = m(G)
Since E is in G, then m(E) <= m(G)
And, $\displaystyle m(G) \leq m(G_n) \leq m(E) + 1/n $
And this is true for all n, thus m(G) <= m(E), and so m(E) = m(G)
and m(G/E) = 0