Take $x_1,\ldots,x_n \in X$ and $y_1,\ldots,y_n\in Y$ and consider the bilinear functional $$B_n(f,g)=\sum_{i\neq j}(i-j)^{-1}f(i)g(j).$$ By Hilbert's inequality, $|B_n(f,g)|\le\pi\|f\|_2\|g\|_2\le\pi n \|f\|_\infty\|g\|_\infty$. On the other hand, $$\sum_{i \neq j}|i-j|^{-1}=2\sum_{k=1}^{n-1}\frac{n-k}{k}\sim2n\log n.$$ So the measure $\mu_n$ representing $B_n$ has norm of order $n\log n$.
Hence, for $\lambda_n>0$ such that $\sum_n \lambda_n n <\infty$ and $\sum_n \lambda_n n\log n=\infty$, one obtains a bounded bilinear form $\sum_n \lambda_n B_n$ which does not extends to a bounded measure. This construction is somewhat canonical. If you want to study it more, look up Grothendieck's inequality.