A decomposition of a measure space with respect to a multiplication operator

Abstract

Let A be a bounded multiplication operator on L 2 ( Ω , m ) {L_2}(\Omega ,m) , where Ω \Omega is a complete separable metric space and m a Borel measure. A set of measure zero can be removed from Ω \Omega so that the multiplicity function of A is equal to the cardinality of the preimage. In the proof, Ω \Omega is decomposed into subsets of simple multiplicity.