A note on the ergodic theorem
The authors give a demonstration of the Birkhoff Ergodic Theorem for ``half-invariant'' measures. They consider a measure space \((X,\mathcal B,\mu)\) and a map \(T:X\to X\) such that the measure \(\mu\) is half \(T\)-invariant, i.e., \(\mu(A)\geq\mu(T^{-1}(A))\), for any \(A\in\mathcal B\). The invariance of the measures allows to estimate frequency of visits to a set \(A\) since almost every point in the phase space is recurrent. Thanks to the half invariance, a composition operator \(\mathcal U_T(\varphi)=\varphi\circ T\) can be defined on \(L^1(X,\mu)\) which is a positive contraction in \(L^1(X,\mu)\). This operator induces on \(X\) a Hopf decomposition. The proof of the theorem is essentially obtained by estimating the points for which the convergence of the Birkhoff averages is too slow.
scientific article; zbMATH DE number 6391424
scientific article