Vague convergence of product measure

Let $$\partial G$$ be the boundary of the $$n$$-ball in $$\mathbb{R}^n$$ and $$[a,b]$$ an interval in $$\mathbb{R}$$.

Let $$\mu$$ be the product measure of the surface measure on $$\partial G$$ and the Lebesgue measure on $$[a,b]$$. I would like to show the vague convergence of a measure $$\mu_N$$ against $$\mu$$.

Let's say I can show for all $$A\subset \partial G$$ and $$B\subset [a,b]$$ the limit $$\lim_{N\rightarrow \infty}\mu_N(A\times B) = \mu(A\times B).$$

My question is which subsets of $$\partial G$$ and $$[a,b]$$ do I actually need to show vague convergence. I know the following for the vague convergence on the interval $$[a,b]$$:

A Sequence $$\{\mu_n,n\geq 1 \}$$ of subprobability measures is said to converge vaguely to an subprobability measure $$\mu$$ if and only there exists a dense subset D of $$[a,b]$$ such that $$\forall c \in D, d\in D, c

Is there a similar condition for the vague convergence on the boundary $$\partial G$$ of the $$n$$-ball in $$\mathbb{R}^n$$?

Let $$\mu_1$$ and $$\mu_2$$ denote the measures on $$\partial G$$ and $$[a,b]$$, respectively, s.t. $$\mu=\mu_1\times\mu_2$$. The sequence of measures $$\mu_N$$ converges weekly to $$\mu$$ iff $$\mu_N(A\times B)\to \mu(A\times B)$$ for each $$\mu_1$$-continuity set $$A$$ and each $$\mu_2$$-continuity set $$B$$. (See, e.g., Theorem 2.7 here.)