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?

Since $\mathbb E[X\mid Y]$ is $\sigma(Y)$-measurable, then $X-\mathbb E[X\mid Y]$ is $\sigma(Y)$-measurable if and only if $X$ is $\sigma(Y)$-measurable.
More generally, if $(S,\mathcal A)$ is a measure space and $f,g\colon S\to\mathbb R$ are maps, $f$ is Borel-measurable but not $g$, then $f+g$ is not Borel-measurable. Otherwise, $f+g-f$ would be Borel-measurable as a sum of two such functions. 
