Krivine's Function Calculus and Bochner Integration

Abstract

We prove that Krivine's Function Calculus is compatible with integration. Let (Omega, Sigma, mu) be a finite measure space, X a Banach lattice, x epsilon X-n, and f : R-n x Omega -> R a function such that f(., w) is continuous and positively homogeneous for every w E 12, and f (s, ") is integrable for every s E R. Put F(s) = f f (s, w) d (w) and define F(x) and f (x, w) via Krivine's Function Calculus. We prove that under certain natural assumptions F(x) = f f (x, w) d (w), where the right hand side is a Bochner integral.