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Hopf's ratio ergodic theorem has an inherent symmetry which we exploit toprovide a simplification of standard proofs of Hopf's and Birkhoff's ergodictheorems. We also present a ratio ergodic theorem for conservativetransformations on a $\sigma$-finite measure space, generalizing Kingman'sergodic theorem for subadditive sequences and generalizing previous results byAkcoglu and Sucheston.

Ratio ergodic theorems: From Hopf to Birkhoff and Kingman