# Finitely additive measures and complementability of Lipschitz-free spaces

## TL;DR

This paper investigates the structure of finitely additive measures on finite-dimensional spaces and demonstrates that Lipschitz-free spaces over such spaces are complemented in their biduals, with specific results for Euclidean spaces.

## Contribution

It proves that Lipschitz-free spaces over finite-dimensional normed spaces are complemented in their biduals, providing new insights into their structure and projections.

## Key findings

- Lipschitz-free space over finite-dimensional space is complemented in its bidual.
- For Euclidean spaces, the projection norm is exactly 1.
- Several structural facts about finitely additive measures on finite-dimensional spaces.

## Abstract

We prove in particular that the Lipschitz-free space over a finitely-dimensional normed space is complemented in its bidual. For Euclidean spaces the norm of the respective projection is $1$. As a tool to obtain the main result we establish several facts on the structure of finitely additive measures on finitely-dimensional spaces.