Let G be a group of measure-preserving transformations on a non-atomic probability measure space \((X,{\mathcal B},\mu)\) (if we are interested in the according topological versions we assume that G is locally compact and the action on the sets \(A\in {\mathcal B}\) is continuous with respect to the metric \(\rho (A,B)=\mu (A\Delta B)).\) The main result is roughly speaking the following: Assume that G acts ergodically, then either there exist ''arbitrary small almost invariant subsets'' or there exist no nontrivial almost invariant subsets; it is even shown that then G acts in a certain strong sense disjointly on small subsets. This can be done by proving zero-one and zero-two laws. The proofs are based mainly on methods from harmonic analysis in connection with the theory if invariant means and representation theory. An essential part of the results can be generalized to the case \(\mu (X)=\infty,\) X locally compact (this case already follows from \textit{V. Losert} [Comment. Math. Helv. 54, 133-139 (1979; Zbl 0396.43007)] and \textit{A. Derighetti} [Comment. Math. Helv. 46, 226-239 (1971; Zbl 0223.43008)] and an application of Namioka's proof of Følner's condition for amenable groups). The situation for \(\mu (X)=1\) is different. We generalize results for countable discrete groups of \textit{K. Schmidt} [Isr. J. Math. 37, 193-208 (1980; Zbl 0485.28018)], \textit{J. Rosenblatt} [Trans. Am. Math. Soc. 265, 623-638 (1981; Zbl 0464.28008)]; \textit{A. Del Junco} and \textit{J. M. Rosenblatt} [Math. Ann. 245, 185-197 (1979; Zbl 0398.28021)] and \textit{V. Losert} and the author [Bull. Lond. Math. Soc. 13, 145-148 (1981; Zbl 0462.43002)] to locally compact groups. For the connection with unique invariant means (in this paper generalized to topological invariant means), we also refer to \textit{A. Connes} and \textit{B. Weiss} [Isr. J. Math. 37, 209-210 (1980; Zbl 0479.28017)], \textit{G. A. Margulis} [Monatsh. Math. 90, 233-236 (1980; Zbl 0425.43001)] and \textit{V. G. Dinfel'd} [Funkts. Anal. Prilozh. 18, No.3, 77 (1984; Zbl 0576.28019)]. Remark: It can be proved that the ''almost independence property'' also discussed in this paper is equivalent to the property \((V=2)\) for connected groups and groups of automorphisms of compact Abelian groups. In particular this property holds for \((Sl(n,{\mathbb{Z}}),({\mathbb{R}}/{\mathbb{Z}})^ n)\) \(n\geq 2,\) or for ergodic actions of semi-simple Lie groups.