  We give an alternative proof of a fact that a finite continuous
non-decreasing submodular set function on a measurable space can be expressed
as a supremum of measures dominated by the function, if there exists a class of
sets which is totally ordered with respect to inclusion and generates the
sigma-algebra of the space. The proof is elementary in the sense that the
measure attaining the supremum in the claim is constructed by a standard
extension theorem of measures. As a consequence, a uniquness of the supremum
attaining measure also follows. A Polish space is an examples of the measurable
space which has a class of totally ordered sets that generates the Borel
sigma-algebra.
