Consider the set of probability measure $M(X\times Y)$ on $X\times Y$ with $X$ and $Y$ finite dimensional spaces (for example euclidean or discrete space).
Consider the application $f:M(X\times Y)\to \mathbb{R}$ linear on $\mu\in M(X\times Y)$.
I know that a linear application on a finite dimensional space is always continuous. Is this correct for $f$ as well? My guess is that the set of probability measure is finite dimensional as well which lead to continuity of the application.
Thank you for your help.