The authors prove the following extension of Furstenberg’s theorem [J. Furstenberg, J. Anal. Math. 31, 204–256 (1977; Zbl 0347.28016)]: Let be a measure-preserving invertible transformation on a probability space and be bounded measurable functions. Then the limit
exists in square mean. This answers a long standing question, on which several authors made partial progress in the last 30 years. The authors also prove an extension where averages are taken over cubes of growing size. The proofs use special types of factors, called characteristic.