# Some Remarks on Capacitary Integrals and Measure Theory

## TL;DR

This paper explores Choquet integrals with minimal assumptions, establishing key properties like sublinearity and convergence theorems, thereby advancing measure theory for non-additive set functions.

## Contribution

It proves equivalence of sublinearity and strong subadditivity without regularity assumptions and extends classical measure convergence theorems to non-additive integrals.

## Key findings

- Sublinearity is equivalent to strong subadditivity for Choquet integrals.
- Standard measure convergence theorems hold for non-additive integrals under minimal assumptions.
- Results generalize classical measure theory to non-additive set functions.

## Abstract

We present results for Choquet integrals with minimal assumptions on the monotone set function through which they are defined. They include the equivalence of sublinearity and strong subadditivity independent of regularity assumptions on the capacity, as well as various forms of standard measure theoretic convergence theorems for these non-additive integrals, e.g. Fatou's lemma and Lebesgue's dominated convergence theorem.