  This paper introduces a notion of decompositions of integral varifolds into
countably many integral varifolds, and the existence of such decomposition of
integral varifolds whose first variation is representable by integration is
established. Furthermore, this result can be generalized by replacing the class
of integral varifolds by some classes of rectifiable varifolds whose density is
uniformly bounded from below. However, such decomposition may fail to be
unique.
