Suppose we have a probability space $X\times Y$ and product measure $\mu=\mu_X\times \mu_Y$.
Given $f\in L^2(X\times Y)$, I can write $f(x,y)=\sum_{j=1}^{d}\sum_{k=1}^da_{j,k}(x)b_{j,k}(y)$ for some $d\in \mathbb{N}$, $a_{j,k}(x)$ non-zero and $b_{j,k}:Y\to \mathbb{R}$. Can I conclude that each $b_{j,k}\in L^2(Y)$?
Is this just a consequence of Fubini's theorem? Or can this be seen by writing $b_{j,k}$ in terms of $f$? If not, are there further conditions one could impose on the $a_{j,k}$? E.g. boundedness... Thanks!
No you cannot. Consider a function $b(y)$ with $\int |b| d\mu_Y=\infty$. Now clearly the function $f(x,y)=0$ is in $L^2(X,Y)$, but we can also write $f$ as $b(y)+b(y)+(-b(y))+(-b(y))$, where the functions $a_{i,j}(x)=1$ are non-negative.
You cannot even conclude that the functions $b_{i,j}$ are measureable, since we could do the same trick for a non-measurable function b.
If we however impose, that the $b's$ are non-negative and measurable, then $a_{i,j}b_{i,j} \leq f$ for all $i,j$, and if $a_{i,j}$ are bounded by a constant $C$ we would have $b_{i,j} \leq \frac{f}{C}$, where $\frac{f}{C}\in L^2.$