The decomposition theorem for functions satisfying the law of large numbers (Q753235)

Let B denote a Banach space and (S,\({\mathcal S},\mu)\) a probability space. The author proves that f: \(S\to B\) satisfies the strong law of large numbers if and only if there exist a Bochner integrable function \(f_ 1\) and a Pettis integrable function \(f_ 2\), \(\| f_ 2\| =0\), where \(\| \cdot \|\) is the Glivenko-Canteli norm, such that \(f=f_ 1+f_ 2\). Moreover the decomposition is unique.