Extensions of Partial Homomorphisms in Probability Theory

Abstract

Two years ago in Amsterdam when trying to solve a problem about invariant measures on semigroups A.A.Balkema and myself arrived to the following very simple question: Are there continuous nontrivial homomorphisms from the convolution semigroup of probability distribution functions F /the topology is the Levy metric/ to the usual additive topological group of the real line R. /A homomorphism is trivial if the image of every element is o./ At that time we could not find the answer to this question. One year later G.Halasz proved that the answer is negative. In fact a stronger result is also true: there is no continuous homomorphism from F to the complex unit circle /the operation is the complex multiplication and the topology is the usual one/ except the trivial homomorphism. In the theory of topological groups these type of continuous homomorphisms /the group representations/ play very important role. The above mentioned result states that there is no continuous representation of the semigroup F. A natural further question is the following: Are there /non-continuous, nontrivial/ homomorphisms from the semigroup F to the group R. Here the answer is affirmative. In [6] we have proved somewhat more: There exists a homomorphism ϕ from F to R such that ϕ(F) = E(F) for every F 6 F having finite expectation E(F). This theorem was implied by the following algebraic LEMMA /for the proof see [6]/.