Convergence in Distribution and Stein’s Method

What should it mean for us to say that two distributions are close, or that a sequence μn of distributions of some random variables Xn taking values in some state space X converges to another distribution μ of some random variable X? Certainly we’ll need to know if Xn and X are close, so we’ll restrict ourselves to state spaces X that are sigma-compact metric spaces (and to avoid needless technical difficulties we’ll take those metric spaces to be complete and separable, or “Polish”— so-called because they were first studied by Sierpiński, Kuratowski, Tarski, and other Polish mathematicians). One approach would be to require that the sequence μn(B) should converge to μ(B) for some class of Borel sets B ⊂ X , or that integrals Eh(Xn) = ∫