Krivine's Function Calculus and Bochner integration

TL;DR

This paper demonstrates that Krivine's Function Calculus can be integrated with Bochner integration, establishing conditions under which these two mathematical frameworks are compatible for functions on Banach lattices.

Contribution

It proves the compatibility of Krivine's Function Calculus with Bochner integration for a class of functions on Banach lattices, under natural assumptions.

Findings

Krivine's Function Calculus is compatible with Bochner integration.

The paper establishes conditions for the interchange of calculus and integration.

Results apply to functions with continuous, positively homogeneous, and integrable slices.

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.