# A positive function with vanishing Lebesgue integral in Zermelo-Fraenkel set theory

## TL;DR

This paper investigates whether the existence of a positive function with zero Lebesgue integral depends on the set-theoretic axioms, particularly within Zermelo-Fraenkel set theory.

## Contribution

It demonstrates how the existence of such functions is influenced by the choice axioms in set theory, especially in models like Feferman-Levy.

## Key findings

- Existence of positive functions with zero Lebesgue integral varies with set-theoretic assumptions.
- In certain models, such functions can exist despite being impossible under standard ZF.
- The results highlight the dependence of measure-theoretic properties on foundational set theory.

## Abstract

Can a positive function on R have zero Lebesgue integral? It depends on how much choice one has. Keywords: Lebesgue integral; Zermelo--Fraenkel theory; Feferman-Levy model