Abstract
Suppose that G is a locally compact abelian group, u an element of infinite order, and w a complex number of modulus 1. It is a familiar fact that there is a complex homomorphism Ψ of the measure algebra M of G, which maps ϵu (the unit mass concentrated at u) to w. Beyond this, one may specify an element μ of M, and require a homomorphism Ψ which does not annihilate μ. The resolution of this problem leads to an abstract lemma on measurable transformations, derived in some generality in the first section.
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