  The goal of this work is to extend the concepts of generalized Lelong number
of positive currents investigated by Skoda, Demailly and Ghiloufi in complex
analysis, to weakly positive supercurrents on the real superspaces. We
generalize then a result of Lagerberg when the supercurrent is closed as well
as a very recent result of Berndtsson for minimal supercurrents associated to
submanifolds of $\mathbb{R}^n$. The main tool is a variant of the well-known
Lelong-Jensen formula in the superformalism case. Moreover, we extend to our
setting various interesting theorems in complex analysis such as Demailly and
Rashkovskii comparison theorems. We also complete the work begun by Lagerberg
on the degree of positive closed supercurrents and we prove a removable
singularities result for positive supercurrents.
