  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.
