# MathSciDoc: An Archive for Mathematician ∫

#### Complex Variables and Complex Analysismathscidoc:1701.08009

Arkiv for Matematik, 51, (1), 125-156, 2010.8
We will introduce a quantity which measures the singularity of a plurisubharmonic function\$φ\$relative to another plurisubharmonic function\$ψ\$, at a point\$a\$. We denote this quantity by\$ν\$_{\$a\$,\$ψ\$}(\$φ\$). It can be seen as a generalization of the classical Lelong number in a natural way: if\$ψ\$=(\$n\$−1)log| ⋅ −\$a\$|, where\$n\$is the dimension of the set where\$φ\$is defined, then\$ν\$_{\$a\$,\$ψ\$}(\$φ\$) coincides with the classical Lelong number of\$φ\$at the point\$a\$. 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\$:\$ν\$_{\$z\$,\$ψ\$}(\$φ\$)≥\$c\$} where\$c\$>0, are in fact analytic sets, provided that the\$weight\$\$ψ\$satisfies some additional conditions.
```@inproceedings{aron2010a,
title={A new generalization of the Lelong number},
author={Aron Lagerberg},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203634614441036},
booktitle={Arkiv for Matematik},
volume={51},
number={1},
pages={125-156},
year={2010},
}
```
Aron Lagerberg. A new generalization of the Lelong number. 2010. Vol. 51. In Arkiv for Matematik. pp.125-156. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203634614441036.