Continuous integration logic

TL;DR

This paper develops a logical framework combining continuous and integral logics for metric measure spaces, proving key theorems like ultraproduct and compactness, with applications in metric measure theory.

Contribution

It introduces a novel logical framework integrating continuous and integral logics tailored for metric measure spaces, establishing foundational theorems and applications.

Findings

Proved the ultraproduct theorem for the new logic

Established compactness and related results in the framework

Applied the results to metric measure theory problems

Abstract

We combine continuous and integral logics and found a logical framework for metric measure spaces equipped with a family of continuous relations and operations. We prove the ultraproduct theorem and deduce compactness and other usual results. We also give applications of the compactness theorem in metric measure theory.