If $\{f_n\}$ is a sequence of µ-measurable functions on $X$ and $\{g_n\}$ is a sequence of $\nu$- measurable functions on $Y$ such that $$\sum\limits_{n=1}^{\infty} f_n(x)g_n(y)$$ converges for all $(x,y) \in X × Y$, prove that this sum defines a function that is $\mu\times\nu$-measurable function on $X \times Y$.
I don't know how to prove a measurable functon in product space. Could you give me a hint? Thanks.
Since sums of measurable functions are measurable and limits of measurable functions are measurable you only have to prove that $f_n(x)g_n(y)$ is measurable for each $n$. Further, products of two measurable functions is measurable so you only have to show that the functions $F(x,y)=f_n(x)$ and $G(x,y)=g_n(y)$ are measurable. Note that $F^{-1}(A)=\{(x,y):f_n(x)\in A\}=f_n^{-1}(A)\times Y$ so this is measurable for any Borel set $A$. Hence $F$ is measurable. Similarly $G$ is measurable.