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Zabreiko's lemma (from P. P. Zabreiko, A theorem for semiadditive functionals, Functional analysis and its applications 3 (1), 1969, 70–72) is not as well known as it deserves to be and I think it fits the bill to some extent, so let me state that result first:
Lemma (Zabreiko, 1969) Let $X$ ...