See this amazing paper ...

Fremlin, David H.; Talagrand, Michel
A decomposition theorem for additive set-functions, with applications to Pettis integrals and ergodic means. Math. Z. 168 (1979), no. 2, 117–142.

http://www.ams.org/mathscinet-getitem?mr=544700

See this amazing paper ...

Fremlin, David H.; Talagrand, Michel
A decomposition theorem for additive set-functions, with applications to Pettis integrals and ergodic means. Math. Z. 168 (1979), no. 2, 117–142. DOI: 10.1007/BF01214191, eudml

https://mathscinet.ams.org/mathscinet-getitem?mr=544700