Laplace's Method Revisited: Weak Convergence of Probability Measures

Let Q be a fixed probability on the Borel -field in Rⁿ and H be an energy function continuous in Rⁿ. A set N is related to H by N = \x ᵧH (y) = H (x) \. Laplace's method, which is interpreted as weak convergence of probabilities, is used to introduce a probability P on N. The general properties of P are studied. When N is a union of smooth compact manifolds and H satisfies some smooth conditions, P can be written in terms of the intrinsic measures on the highest dimensional mainfolds in N.