Uniqueness of invariant means for measure-preserving transformations

Author:
Joseph Rosenblatt

Journal:
Trans. Amer. Math. Soc. **265** (1981), 623-636

MSC:
Primary 28D15; Secondary 43A07, 58F11

MathSciNet review:
610970

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Abstract: For some compact abelian groups (e.g. , , and ), the group of topological automorphisms of has the Haar integral as the unique -invariant mean on . This gives a new characterization of Lebesgue measure on the bounded Lebesgue measurable subsets of , ; it is the unique normalized positive finitely-additive measure on which is invariant under isometries and the transformation of . Other examples of, as well as necessary and sufficient conditions for, the uniqueness of a mean on , which is invariant by some group of measure-preserving transformations of the probability space , are described.

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DOI:
https://doi.org/10.1090/S0002-9947-1981-0610970-7

Article copyright:
© Copyright 1981
American Mathematical Society