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, $x\in X^n$, and $f\colon\mathbb R^n\times\Omega\to\mathbb R$ a function such that $f(\cdot,\omega)$ is continuous and positively homogeneous for every $\omega\in\Omega$, and $f(s,\cdot)$ is integrable for every $s\in\mathbb R^n$. Put $F(s)=\int f(s,\omega)d\mu(\omega)$ and define $F(x)$ and $f(x,\omega)$ via Krivine's Function Calculus. We prove that under certain natural assumptions $F(x)=\int f(x,\omega)d\mu(\omega)$, where the right hand side is a Bochner integral.