Sum of measurable and non-measurable functions The context of my question is probabilistic. The Doob-Dynkin lemma asserts that a r.v $X$ is $\sigma (Y)$ measurable iff it's a deterministic function of $Y$. Now, the conditional expectation $\mathbb{E} (X\mid Y)$ may be viewed as the best approximation of $X$ using deterministic functions of $Y$. I want to know whether $X-\mathbb{E} (X\mid Y)$ is necessarily not a function of $Y$, in other words, not $\sigma (Y)$ measurable.
More generally, is the sum of a measurable function with a non-measurable function generally non-measurable?

I think it is always a non measurable function.
let $f$ be the measurable function and let g be non measurable then if $f+g$ is measurable say $h$. Then $h-f=g$ is measurable, a contradiction. 
