# Decomposability, Convexity and Continuous Linear Operators in $L^1(\mu,E)$: The Case for Saturated Measure Spaces

## TL;DR

This paper extends the convexity properties of integrals over decomposable sets in Banach spaces by introducing the concept of saturation in measure spaces, providing a comprehensive characterization of decomposability.

## Contribution

It introduces the notion of saturation in measure spaces and links it to the convexity of integrals in Banach spaces, generalizing the Lyapunov convexity theorem.

## Key findings

- Convexity of the integral of decomposable sets holds under saturation.
- Saturation characterizes decomposability in measure spaces.
- Extension of Lyapunov's theorem to infinite-dimensional spaces.

## Abstract

Motivated by the Lyapunov convexity theorem in infinite dimensions, we extend the convexity of the integral of a decomposable set to separable Banach spaces under the strengthened notion of nonatomicity of measure spaces, called "saturation", and provide a complete characterization of decomposability in terms of saturation.