Pluripolar set

In mathematics, in the area of potential theory, a pluripolar set is the analog of a polar set for plurisubharmonic functions.

Definition
Let $$G \subset {\mathbb{C}}^n$$ and let $$f \colon G \to {\mathbb{R}} \cup \{ - \infty \}$$ be a plurisubharmonic function which is not identically $$-\infty$$. The set


 * $${\mathcal{P}} := \{ z \in G \mid f(z) = - \infty \}$$

is called a complete pluripolar set. A pluripolar set is any subset of a complete pluripolar set. Pluripolar sets are of Hausdorff dimension at most $$2n-2$$ and have zero Lebesgue measure.

If $$f$$ is a holomorphic function then $$\log | f |$$ is a plurisubharmonic function. The zero set of $$f$$ is then a pluripolar set.