Sorry if this turns out to be a silly question, but I am having difficulties in both understanding it and finding other references for it. I hope that someone can clear my concepts here on overflow.


Recently, I am reading the book "Degenerate Complex Monge-Ampère Equations" by Guedj and Zeriahi and having some confusion which I cannot find any references for.


In chapter 1, section 1.2.3, it is stated that if $u\in SH(\Omega)$, where $SH(\Omega)$ denotes the set of subharmonic functions over a domain $\Omega\subset\mathbb C$, then the *distribution* $\Delta u\geq 0$ is a nonnegative distribution. My first question is that why is the laplacian of $u$ a distribution. (From my understanding, a distribution is supposed to be a continuous linear functional from the space of compactly supported smooth functions to $\mathbb C$. Then I do not understand why is $\Delta u$ a distribution.)


On the next page, the authors further define the Riesz measure of $u\in SH(\Omega)$ by $\mu\_u:=\frac{1}{2\pi}\Delta u$. I am not sure how does it become a measure. (From my understanding, a measure is supposed to act on subsets of $\Omega$, then I do not understand how $\Delta u$ can be regarded as a measure.)


I would be very grateful if someone could explain the missing linkages between them or could provide me with some references on possibly the theory of distribution. Thank you.



How to understand subharmonic functions, distributions, and measure?
Before reading such advanced books on "degenerate Monge Ampere equations", you need a background in subharmonic functions and potential theory (and distributions if you do not have it). Some good introductory books are Notions of Convexity by L. Hormander,
and Subhrmonic Functions, v. 1, by W. Hayman and P. Kennedy.


