On an inequality of A.~Grothendieck concerning operators on $L^1$

In 1955, A.~Grothendieck proved a basic inequality which shows that any bounded linear operator between $L^1(\mu)$-spaces maps (Lebesgue-) dominated sequences to dominated sequences. An elementary proof of this inequality is obtained via a new decomposition principle for the lattice of measurable functions. An exposition is also given of the M.~L\'evy extension theorem for operators defined on subspaces of $L^1(\mu)$-spaces.