Pointwise CDF convergence to a continuous law is uniform

The stated replacement on each measurable null set preserves measurable evaluations and their pointwise convergence in probability, and gives CDF paths everywhere. Consequently ${\sup }_x\mathrm{\mid }{G}_n(x)-F(x)\mathrm{\mid }$ equals the supremum over rational $x$ for every outcome, by right-continuity, and is measurable. Fix $0<\eta <1\mathrm{/}2$. The events defining a CDF increase to the whole space as $x\to +\mathrm{\infty }$ and decrease to the empty set as $x\to -\mathrm{\infty }$; [F3]–[F4] therefore give its tail limits. Choose $L<R$ with $F(L)<\eta$ and $1-F(R)<\eta$. By [F2], $F$ is uniformly continuous on $[L,R]$, so choose a finite partition $L=x_0<\cdots <x_k=R$ for which every increment $F(x_{j+1})-F(x_j)<\eta$.

Pointwise convergence at the finitely many grid points implies that, with probability tending to one, $\mathrm{\mid }{G}_n(x_j)-F(x_j)\mathrm{\mid }\le \eta$ for every $j$. On this event, monotonicity of both CDFs bounds their difference on each $[x_j,x_{j+1}]$ by $3\eta$. For $x<L$, both CDF values lie between $0$ and their values at $L$, so their difference is at most $2\eta$; for $x>R$, apply the same argument to the complementary tails using their values at $R$. Hence ${\sup }_x\mathrm{\mid }{G}_n(x)-F(x)\mathrm{\mid }\le 3\eta$ with probability tending to one. Since $\eta >0$ was arbitrary, the supremum converges to zero in probability.