Deducing measurability of $f$ and $g$ from the measurability of $f+g$?

Question

I was thinking of this - If $f$, $g$ are measurable function then $f+g$ is measurable. (For example, see this post: Proving that sum of two measurable functions is measurable.)

But what about the converse that is if $f+g$ is measurable then I think it may not be true that $f$ and $g$ will be measurable.

Is this correct? Any counterexample we can give?

Answer 1

Let $f$ be non-measurable and let $g=−f$. Then $f+g=0$ which is measurable and none of $f,g$ is measurable.