An Application of a Pisier Factorization Theorem to the Pettis Integral

There are several generalizations of the space L1(R) of Lebesgue integrable functions taking values in the real numbers R (and defined on the usual Lebesgue measure space (Ω,Σ, μ) on [0, 1] ) to a space of strongly-measurable “integrable” (suitably formulated) functions taking values in a Banach space X. The most common generalization is the space L1(X) of Bochner-Lebesgue integrable functions. Using the fact [P1, Theorem 1.1] that a strongly-measurable function is essentially separably-valued, one can easily extend Lebesgue’s Differentiation Theorem from L1(R) to L1(X). Specifically [B; cf. DU, Theorem II.2.9], if f ∈ L1(X), then lim h→0 1 h ∫ t+h