Suppose that $S$ is a separable metric space or Polish. Let $μ_{n},n∈N $ be a random probability measures and let μ be a deterministic probability measure on $S$. That is to say, that the $ μ_{n}$ are measurable maps from a probability space $(Ω,T,P)$ to the space of $M_1(S) $ equipped with the Borel-σ-algebra generated by the topology of weak convergence. Assume that the expected measures $ν_{n}:=Eμ_{n}$, defined via duality as $∫f dν_{n}:=E∫f dμ_{n}$ for all $f∈C_{b}(S)$, converge weakly to μ, i.e. that for all $f∈C_{b}(S)$ we have the convergence of $E∫fdμ_{n}$ to $∫fdμ$.

For all ϵ>0 the sequence $P(d_{BL}(μ_{n},μ)>ϵ)$ converges to zero, where $d_{BL}$ is the bounded Lipschitz metric $$ d_{BL}(μ,ν)=\sup\Big\{\Big\vert∫f dμ−∫f dν\Big|: f:S\to\mathbb{R} \; \mbox{is 1-Lipschitz and $\Vert f\Vert_\infty\leq 1$}\Big\} $$ which completely metrizes the topology of weak convergence)

For all $f∈C_{b}(S)$ it holds that $∫f dμ_{n}$ converges in probability to $∫f dμ$.

please i want to know if 1) implies 2) or if 2) implies 1) or if 1) is equivalent to 2)

and in the case where $μ_{n}$ is not necessary a probability measure but just a random measure, do the same results remain true?