# Measure extension by local approximation

## TL;DR

This paper introduces a measure extension theorem based on local approximation of sets by algebra elements, establishing a connection with Carathéodory-measurability.

## Contribution

It provides a new measure extension theorem using local approximation and characterizes Carathéodory-measurable sets through local approximability.

## Key findings

- Proves a measure extension theorem via local approximation.
- Shows equivalence between local approximability and Carathéodory-measurability.
- Establishes a new criterion for measurability based on local approximation.

## Abstract

Measurable sets are defined as those locally approximable, in a certain sense, by sets in the given algebra (or ring). A corresponding measure extension theorem is proved. It is also shown that a set is locally approximable in the mentioned sense if and only if it is Carath\'eodory-measurable.