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.