# On additive property of finitely additive measures

## TL;DR

This paper generalizes the conditions under which countable sums of finitely additive measures possess the additive property, extending previous finite sum results and applying them to density measures.

## Contribution

It extends Basile and Rao's finite sum condition to countable sums of finitely additive measures and explores applications to density measures.

## Key findings

- Generalized additive property condition to countable sums
- Extended finite sum results to infinite sums
- Applied findings to density measures

## Abstract

By the additive property, we mean a condition under which $L^p$ spaces over finitely additive measures are complete. Basile and Rao gives a necessary and sufficient condition that a finite sum of finitely additive measures has the additive property. We generalize this result to the case of a countable sum of finitely additive meaures. An application of this result to density measures are also presented.