Some remarks on measures on noncompact semisimple Lie groups

Abstract

This paper answers a question posed by K. R. Parthasarathy: Let X be a symmetric space of non-compact type and G the connected component of the group of isometries of X. Let m be the canonical G-invariant measure on X and E a Borel set in X such that E is compact and 0 < m(E) < ∞. If μ,ν are probability measures on X such that μ(g ⋅ E) = ν(g ⋅ E) for all g ∈ G, then is μ = ν? We answer the question in the affirmative (Theorem A) and also find that the condition “E is compact” is unnecessary. A special case of this problem (under the condition that μ and ν are K-invariant probabilities on X, where K is a maximal compact subgroup of G) was settled by I. K. Rana.