If a set $E$ differs from a measurable set by a set of measure zero, then $E$ is measurable

Question

Suppose that $F$ is Lebesgue measurable and $m(E-F)=0$. I want to convince myself that $E$ is also measurable, which seems intuitively true.

Since $F$ and $E-F$ are measurable, by additivity we have $m(E\cup F)=m(E-F\cup F)=m(E-F)+m(F)=m(F)$

Am I missing something obvious?