On the Pointwise Ergodic Theorem in ${\displaystyle L_p}$

Let (X, F, μ) be a finite measure space, and φ an invertible, nonsingular transformation on (X, F, μ). We prove that the Pointwise Ergodic Theorem (P.E.T.) in ${\displaystyle L_p\left(d\mu \right)}$ holds for the operator ${\displaystyle Tf=f\circ \varphi }$ if, and only if, it holds for the formal adjoint of T in ${\displaystyle L_q\left(d\mu \right)(1/p+1/q=1)}$. We also characterize the P.E.T. in terms of the Radon-Nikodym derivative of the measure μ with respect to an invariant measure.