Bernstein's theorem states that for any completely monotone function $f$: $f \in C^{\infty}[0,+\infty)$, $(-1)^n f^{(n)}(t) \geqslant 0$ there is a finite Borel measure $\mu$ such that 
$$ f(t) = \int\_{0}^{+\infty} e^{-tx} \mu(dx) $$


Is there some generalisation of this result on the case of $n$ dimensions?



On the generalisation of Bernstein's theorem on monotone functions
Yes, there is and it goes by the name of Bochner theorem. For details see the book of A. Klenke: Probability Theory. A Comprehensive Course, Springer Verlag, 2008. 


