A new generalization of the Lelong number

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We introduce a quantity which measures the singularity of a plurisubharmonic function f relative to another plurisubharmonic function g, at a point a. This quantity, which we denote by νₐ,g(f), can be seen as a generalization of the classical Lelong number, in a natural way. The main theorem of this article says that the upper level sets of our generalized Lelong number, i.e. the sets of the form {z: νz,g(f) ≥ c > 0 }, are in fact analytic sets, under certain conditions on the weight g.