Open-Invariant Measures and the Covering Number of Sets

A result of J. Mycielski says that on every metric space (X, ?) with a non-empty compact thick set C ? X there exists a regular open-invariant Borel measure μ with μ(C) = 1. μ is called open-invariant if μ(A) = μ(B) for open isometric sets A, B ? X. We relate this result to the notion of a Hewitt-Stromberg measure and give a new independent existence proof for such an open-invariant measure μ on a compact metric space (X, ?). This proof works by induction, the well-known metric outer measure construction of Caratheodory-Hausdorff and a new property of the covering number N(X, q) of X.