An ergodic maximal equality for nonsingular flows and applications (Q1332965)

Let \(\{T_ t,\;t\in \mathbb{R}\}\) be a measurable flow of nonsingular transformations of a probability space \((X,B,\mu)\), and let \(w_ t= d(\mu\circ T_ t)/d\mu\). For \(f\in L_ 1(\mu)\), let \(f^*\) denote the maximal function \[ f^*(x)= \sup_{ b>0} \bigl\{\int^ b_ 0 f\circ T_ t(x) w_ t(x) dt\bigl/ \int^ b_ 0 w_ t(x) dt\bigr\}. \] The main result asserts that \[ \alpha\mu(f^*> \alpha)= \int I(f^*> \alpha)f d\mu \] holds for \(\alpha> \int f d\mu\), where \(I\) denotes the indicator function. (This was shown by Marcus and Petersen in the measure preserving case).