Vol. 110, No. 2, 1984

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Some remarks on measures on noncompact semisimple Lie groups

Alladi Sitaram

Vol. 110 (1984), No. 2, 429–434

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.

Mathematical Subject Classification 2000
Primary: 43A85
Secondary: 22E30
Received: 5 May 1982
Revised: 21 December 1982
Published: 1 February 1984
Alladi Sitaram