  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.
