Stochastic Finance, 4th Edition

A.6 Spaces of measures

Let S be a topological space. S is called metrizable if there exists a metric d on S which generates the topology of S. That is, the open d-balls

Bε(x) := {y ∈ S | d(x, y) < ε}, x ∈ S, ε > 0,

form a base for the topology of S in the sense that a set U ⊂ S is open if and only if it can be written as a union of such d-balls. A convenient feature of metrizable spaces is that their topological properties can be characterized via convergent sequences. For instance, a subset A of the metrizable space S is closed if and only if for every convergent sequence in A its limit point is also contained in A. Moreover, a function f : S → ℝ is continuous at y ∈ S if and only if f (yn) converges to f (y) for every sequence (yn) converging ...