A construction of finite and 𝜎-finite invariant measures in measure spaces

Abstract

Let T T be a bijective nonsingular transformation on a finite measure space. We shall first construct a σ \sigma -finite and finite invariant measure by a unified method which is valid for both cases. Secondly we shall give another construction of a finite invariant measure. We shall also give a new necessary and sufficient condition of a unified form for the existence of σ \sigma -finite and finite invariant measures. Further, we shall discuss in detail ergodic transformations.