Uniform distribution, invariant means, and Riemann integrals (Q1905288)

The author studies actions of semigroups \(S\) satisfying certain amenability conditions. He proves a local version of Dixmier's criterion, looking for invariant means on closed invariant subspaces of \(m(S)\) (the space of complex valued bounded functions on \(S\) generated by a single \(f \in m(S))\) (existence and uniqueness). He also proves some results on the representation of \(S\) on Banach spaces \(E\), considering also the dual representation \(\sigma\) on the dual space \(E'\). In the second part a general concept of uniform distribution is studied: A mapping \(\omega : S \to X\), \((X \mu)\) a compact probability space, is called \(M\)-uniformly distributed (this concept is even generalized to not necessarily positive weights), for certain subsets \(M\) of the set of all \(S\)-left invariant means on \(m(S) \), if \(I(f \circ \omega) = \mu (f)\) for all continuous \(f\) on \(X\). For \(S = N\) (and more generally for left amenable semigroups) it is easy to find subsets \(M\), such that this concept coincides with the usual one (of Cesaro convergence of \(f(x_n)\) if \(S = N)\). Strengthening a result of \textit{E. Yu. Terekhina} on the substitution of variables in Riemann integrals [Mosc. Univ. Math. Bull. 40, 75-79 (1985); translation from Vestn. Mosk. Univ., Ser. I 1985, No. 3, 78-80 (1985; Zbl 0594.26006)] the author shows that there exist certain ``distinctly distributed'' sequences. Finally a result on Riemann integrable functions on compact measure spaces is proved which is related to the variable-substitution problem.