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- September 13, 2020
- By menge

Let Fn be a sequence of sets in F with P(Fn) > 0 and Fn ↓ ∅. Define T0(1E) = 1 if there exists an n such that Fn ⊂ E a.e., and T0(1E) = 0 otherwise. Extend T0 to the simple functions by linearity and check that it is continuous, non-negative, and that ν0(E) := T0(1E) is finitely additive. By the Hahn-Banach Theorem, T0 has a continuous extension, T, to all of L∞. By construction ν(E) := T(1E) is a purely finitely additive probability since ν(Fn) ≡ 1 but Fn ↓ ∅.