Decomposition of positive definite functions defined on a neighbourhood of the identity (Q580626)

Let V be a symmetric open neighbourhood of the identity of a topological group G. We show that every positive definite function f on v can be written as \(f=f_ c+f_ s\) where \(f_ c\) and \(f_ s\) are positive definite functions on V, \(f_ c\) is continuous and \(f_ s\) averages to zero. If G is locally compact with Haar measure \(m_ G\) and f is \(m_ G\)-measurable then \(f_ s=0\), \(m_ G\)-almost everywhere.