Let ${ℛ}_{\alpha }$ be the Riesz distribution on a simple Euclidean Jordan algebra, parametrized by $\alpha \in ℂ$. I give an elementary proof of the necessary and sufficient condition for ${ℛ}_{\alpha }$ to be a locally finite complex measure (= complex Radon measure).

We show that, as in the linear case, the normalized Haar measure on a compact topological group G is a Pietsch measure for nonlinear summing mappings on closed translation invariant subspaces of C(G). This answers a question posed to the authors by J. Diestel. We also show that our result applies to several well-studied classes of nonlinear summing mappings. In the final section some problems are proposed.

For measures on a Cantor space, the demand that the measure be "good" is a useful homogeneity condition. We examine the question of when a Bernoulli measure on the sequence space for an alphabet of size n is good. Complete answers are given for the n = 2 cases and the rational cases. Partial results are obtained for the general cases.