# pr.probability – How does one define weak convergence of probability measures in \$L^{infty}(Omega)\$?

I am reading the following article and on page 9/17 (above Eqn (4.9)) the authors state that if $$gamma_{epsilon_k}|_G_{delta}times Omegato gamma|_G_{delta}times Omega$$ as $$epsilon_kto 0$$ in the weak* limit then $$P^{t}_{#}gamma_{epsilon_k}|_G_{delta}times Omega$$ converges weakly in $$L^{infty}(Omega)$$ to $$P^{t}_{#}gamma|_G_{delta}times Omega,$$ where

• $$Omega$$ is a compact subset of $$mathbb{R}^d,$$
• $$G_{delta}$$ is some borel subset of $$Omega,$$
• $$gamma, gamma_{epsilon_k}in mathcal{P}(Omega times Omega)$$ with first marginal $$pi^{1}_{#} gamma_{epsilon_k} =pi^{1}_{#} gamma =mu << mathcal{L}^{d}$$ and,
• $$P^{t}(x,y)=(1-t)x+ ty$$ for $$tin (0,1)$$ and for all $$x,yin Omega.$$

I am not sure how one defines convergence in $$L^{infty}(Omega)$$ of the push-forward measure $$P^{t}_{#}gamma_{epsilon_k}|_G_{delta}times Omega.$$ Reading the paper so far I am guessing that for any borel set $$Bsubset Omega$$ and $$muin mathcal{P}(Omega)$$ we define the $$L^{infty}$$ norm of the measure as,
$$mu(B)leq ||mu||_{L^{infty}(Omega)} mathcal{L}^{d}(B),$$
but I am not sure. Any references will be much appreciated.