Theorem 21.11.552DC9BC6F295A

Qualification and proof. The topology on $\mathcal P(N_{\mathbb R},P)$ is pointwise convergence. Since all these functions are finite and convex, local uniform convergence of convex functions shows that $F_j\to F$ uniformly on compact subsets. Alexandrov-measure continuity now gives convergence against every compactly supported continuous test function, that is, weak convergence as Radon measures on this noncompact space.

Convergence against all bounded continuous functions would require an additional no-loss-of-mass condition and is not automatic here. For example, on $\mathbb R$ the functions $F_j(x)=\log(1+\exp(x-j))$ have common growth body $[0,1]$ and converge locally uniformly to $0$, while their real Monge--Ampère measures have total mass $1$ and move to $+\infty$. Their vague limit is zero. Thus the weak topology in this assertion must be read as the compactly supported test-function topology; if total masses also converge, a compact-cutoff argument upgrades it to the bounded-continuous topology.