Union of a null set and a non-measurable set

Question

Suppose $S$ is a non-measurable set (wrt the Lebesgue measure) and $N$ has measure 0. What can be said about $S\cup N$? Is it also non-measurable?

Answer 1

Suppose that $S \cup N$ is measurable.

Since $N$ is measurable, it follows that $N^c$ is measurable.

Therefore $(S \cup N) \cap N^c = S \backslash N$ is measurable.

Also, since $N$ has measure zero and since $S \cap N \subset N$, it follows by completeness of Lebesgue measure that $S \cap N$ is measurable.

Therefore $S = (S \cap N) \cup (S \backslash N)$ is measurable.

We showed that if $S \cup N$ is measurable, then $S$ is measurable. By contrapositive, if $S$ is non-measurable, then $S \cup N$ is non-measurable.