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 $f:{\mathbb{R}}^n\times \Omega \to \mathbb{R}$ 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.