Finite-projection convergence and posterior tail control imply weak BvM

For any probability measure $Q$ on $M_0(w)$ and any bounded $1$-Lipschitz function $h$, $\mathrm{\mid }h(x)-h(P_{J}x)\mathrm{\mid }\le \min (2,\mathrm{\parallel }x-P_{J}x{\mathrm{\parallel }}_{M(w)})$. Splitting at $\epsilon >0$ and taking the supremum over such $h$ gives ${\beta }_{BL}(Q,Q\circ P_J^{-1})\le \epsilon +2Q(\mathrm{\parallel }x-P_{J}x{\mathrm{\parallel }}_{M(w)}>\epsilon )$. Applying this to $Q_n$ and $G$, then using the triangle inequality, bounds ${\beta }_{BL}(Q_n,G)$ by the two projection errors and the fixed-$J$ distance between projected laws.

For each $x\in M_0(w)$, [F2] gives $\mathrm{\parallel }x-P_{J}x{\mathrm{\parallel }}_{M(w)}\to 0$. Thus the events $\{x:\mathrm{\parallel }x-P_{J}x{\mathrm{\parallel }}_{M(w)}>\epsilon \}$ decrease to the empty set, and [F3] makes their $G$-probabilities tend to zero. Given any target error, first choose $\epsilon$ small, then choose $J$ large so this $G$-tail and the limiting posterior-tail probability are small; for that fixed $J$, the projected-law distance tends to zero in probability by assumption. Step 1.1 then implies ${\beta }_{BL}(Q_n,G)\to 0$ in probability.