When the integral is the limit
A sequence \((\mu_n)\) of probability measures on \(\mathbb{Z}\) is a \textit{norm good averaging method} if \[ \lim_{n\to\infty}\sum^\infty_{k=-\infty}\mu_n(k) f\circ\tau^k \] exists in \(L^2_m\) for all invertible measure-preserving transformations \(\tau\) of a probability space \((X,{\mathcal B},m)\) and for all \(f\in L^2_m\). A transformation \(\tau\) on \([0,1]\) leaving Lebesgue measure \(\lambda\) invariant (this class is denoted by \(\mathcal T\)) is \textit{good} if this limit equals \(\int f d\lambda\) for all \(f\in L^2_\lambda\). An averaging method is \textit{uniformly dissipative} if \(\lim_{n\to\infty}\sup_{k\in\mathbb{Z}} \mu_n(k)=0\). The following results are proved: 1) Given a uniformly dissipative, norm good averaging method, the set of good transformations of \([0,1]\) is a dense \(G_\delta\)-set in the weak topology on \(\mathcal T\). 2) The set of transformations in \(\mathcal T\) which are good for all uniformly dissipative, norm good averaging methods are of first category. This continues the author's work from Can. J. Math. 46, No. 1, 184-199 (1994; Zbl 0796.28011).
scientific article; zbMATH DE number 1005676
scientific article