I am a bit confused by this Theorem in the book 'Optimal Transport for Applied Mathematicians' of Santambrogio. This question concerns absolutely continuous (w.r.t Lebesgue) probability measures on $\mathbb{R}^d$.
The theorem says that given a sequence $\{\rho_n\}$ of weakly converging probability densities (i.e convergence against $C_b(\mathbb{R}^d)$ functions), if the sequence is equi-integrable (which I think people also call uniformly integrable) defined as equi-integrable definition, then the sequence weakly converges (up to a subsequence) in $L_1(\mathbb{R}^d)$ Theorem equi-integrable implies L1 convergence.
Why I'm confused : I don't understand why we need the weak convergence in the statement of the theorem. Surely : equi-integrability implies tightness, then Prokhorov theorem implies the weak convergence?
What am I missing? How is this strategy different to obtianing convergence via tightness + Prokhorov theorem?