Geodesic
The relation between lebesgue integrable and Birkhoff–Khinchin summable functions
Matematičeskie zametki, Tome 10 (1971) no. 2, pp. 219-228 
V. S. Shul'man. The relation between lebesgue integrable and Birkhoff–Khinchin summable functions. Matematičeskie zametki, Tome 10 (1971) no. 2, pp. 219-228. http://geodesic.mathdoc.fr/item/MZM_1971_10_2_a11/
@article{MZM_1971_10_2_a11,
     author = {V. S. Shul'man},
     title = {The relation between lebesgue integrable and {Birkhoff{\textendash}Khinchin} summable functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {219--228},
     year = {1971},
     volume = {10},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_10_2_a11/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

A necessary and sufficient condition is obtained for a transformation to satisfy the converse to the ergodic theorem.