Let $f(x,y)$ be nonnegative Borel function on $\mathbb{R^2}$.Prove that $\psi(x) = \int_{\mathbb{R}}f(x,y)d\mu(y) $ is also Borel function($\mu $ is symbol for Lebesgue measure).
My attempt:
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$f(x,y) \geq 0$ so there is sequence of positive sample functions $(s_n) $ such that $s_n(x,y) \rightarrow f(x,y) $ as $ n \rightarrow +\infty$.Now I can use Lebesgue Monotone theorem $\lim_{n \to \infty} \int _{\mathbb{R}}s_n(x,y)d\mu(y) = \psi(x)$.If i show that $\int _{\mathbb{R}}s_n(x,y)d\mu(y)$is measurable $ \forall n \in \mathbb{N}$ i am done but i don't know how to do that.
Any answer or hint would be appreciated.
This is part of the proof of Tonnelli's Theorem. You can find a proof in any book which has Fubini/Tonelly Theorem.