Prove weak convergence of probability measures

Question

Let $Q \in P(P(\Sigma))$ be given, whereby $\Sigma$ is a Polish space (complete and separable metric space) and $P(\Sigma)$ denotes the space of probability measures on $\Sigma$ and $P(P(\Sigma))$ the space of probability measures on $P(\Sigma).$ One defines $Q_n \in P(P(\Sigma))$ to be the distribution of $$\alpha = (\alpha_1,...,\alpha_n) \in P(\Sigma)^n \mapsto \frac{1}{n} \sum_{k=1}^n \alpha_k \in P(\Sigma)$$ under $Q^n \in P(P(\Sigma)^n).$ Let $\mu_Q(\Gamma) = \int_{P(\Sigma)} \alpha(\Gamma)Q(d\alpha)\in P(\Sigma), \Gamma \in B_{\Sigma}.$ I need to prove that $Q_n$ converges weakly to $\delta_{\mu_Q}.$ I am confused with the structure of the space $P(P(\Sigma)).$ I have been so far familiar with random variables. In the context above one uses probability measures instead. I need to use the weak law of large numbers (WLLN) given the finite normalized sum (empirical mean) of probability measures. In case of random variables this will be convergence in probability which implies convergence in distribution. In the case of probability measure space and the general setting of weak topology, I am not sure how to use WLLN and the second countability of the weak topology on $P(\Sigma)$ in order to prove the required weak convergence. Can somebody help ? Thanks.