Rectifiable varifolds are a generalization of rectifiable sets in the sense that they allow for a density function to be defined on the set. They are also strictly connected to rectifiable currents, in fact to such a current one can always associate a varifold by putting aside the orientation.
Rectifiable varifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rectifiable_varifold&oldid=27881