Any rule of thumb that says any reasonable function I can write down is measurable?

Question

Question arose in the context of probability: If $Y$ is $F-$measurable and $h$ is some function, $h(Y)$ is $F-$measurable if $h$ is a measurable function (Doob-Dynkin).

For example, let $f(x,y)$ be the joint density function of random variables $X$ and $Y$, $g(x)$ be some integrable function, and $h(y)=\int g(x)f(x,y)\,dx/\int f(x,y)\,dx$; then $h(y)$ is measurable - I wondered if there is some rule of thumb that tells me this quickly, since I remembered someone saying, "Any function you can write down is measurable."

Thanks!

When one says "Any functions you can write down is measurable", he often refers to the fact that all functions being Lebesgue-measurable is consistent with Zermelo-Fraenkel (without the axiom choice). Of course, this needs not hold for the general definition of measurable function between arbitrary mensural spaces. — , Jan 30 '16 at 14:10
There is a theorem that says any constructive function is continuous. (There is a feature of the constructive reals which keeps you from defining functions piecewise.) — Thomas Andrews, Jan 30 '16 at 14:10
@ThomasAndrews: One doesn't have to go so far to full constructivism, otherwise as you note we don't even have piecewise functions, which we certainly know how to write down! — user21820, Jan 30 '16 at 14:16
Which is why I wrote a comment, not an answer. @user21820 But it means that if you write something which is piecewise something computable, then it is piecewise continuous and hence measurable. — Thomas Andrews, Jan 30 '16 at 14:18
@ThomasAndrews: What you say implies that computable implies piecewise continuous, presumably with countably many pieces. Is that really so? — user21820, Jan 30 '16 at 14:19
that's not what I wrote. There is no computable discontinuous on the reals. If a function is piecewise computable, then it is measurable. @user21820 — Thomas Andrews, Jan 30 '16 at 14:20
Oh I see. I forgot that; I'd a different idea of "computable", which wasn't the right one. — user21820, Jan 30 '16 at 14:23

Any rule of thumb that says any reasonable function I can write down is measurable?

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