# Rediscovered theorem of Luzin

## TL;DR

This paper revisits Luzin's 1934 theorem showing every real set can be decomposed into two full subsets, and aims to clarify and prove his original reasoning behind this significant measure-theoretic result.

## Contribution

The paper provides a detailed analysis and proof of Luzin's decomposition theorem, elucidating his original ideas from 1934.

## Key findings

- Confirmed Luzin's decomposition theorem for sets of real numbers.
- Clarified Luzin's original reasoning and methodology.
- Extended understanding of measure and category decompositions.

## Abstract

In 1934 N. N. Luzin proved in his short (but dense) paper \textit{Sur la decomposition des ensembles} that every set $X\subseteq \mathbb{R}$ can be decomposed into two full, with respect to Lebesgue measure or category, subsets. We will try to (at least partially) decipher the reasoning of Luzin and prove this result following his idea.