If two probabilities $P,P'$ on $\mathbb{R}^n$ satistfy $P_\phi$=$P'_\phi$ for all linear $\phi:\mathbb{R}^n \rightarrow \mathbb{R}$ then they are equal. (Here $P_{\phi}(A)=P(\phi^{-1}(A))$)
I know this is true (saw it in probability class a while ago). The proof used characteristic functions. I was very unhappy with it because it was completely unconstructive, with the usual stupid trick saying that two functions are equal iff their Fourier transforms are.
Does somebody know a constructive proof ? I.e, if I give all the $P_\phi$, is there a method to obtain for example $P(x\geqslant x_0, y\geqslant y_0)$ ?