A - Distances on measures

Summary

The properties of separability, metrizability, compactness and completeness for a topological space S are crucial for the analysis of S-valued random processes. Here we shall recall the basis relevant notions for the space of Borel measures, highlighting the main ideas and examples and omitting lengthy proofs.

Recall that a topological (e.g. metric) space is called separable if it contains a countable dense subset. It is useful to have in mind that separability is a topological property, unlike, say, completeness, which depends on the choice of distance. (For example, an open interval and the line R are homeomorphic, but the usual distance is complete for the line and not complete for the interval). The following standard examples show that separability cannot necessarily be assumed.

Example A.1 The Banach space l_∞ of bounded sequences of real (or complex) numbers a = (a₁, a₂, …) equipped with the sup norm ∥a∥ = sup_i∣a_i∣ is not separable, because its subset of sequences with values in {0, 1} is not countable but the distance between any two such (not coinciding) sequences is 1.

Example A.2 The Banach spaces C(R^d), L_∞(R^d), M^signed(R^d) are not separable because they contain a subspace isomorphic to l_∞.

Example A.3 The Banach spaces C_∞(R^d), L_p(R^d), p ∈ [1,∞), are separable; this follows from the Stone–Weierstrass theorem.