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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.

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    $\begingroup$ If you think of $u$ as a distribution (any continuous function is a distribution) then it makes sense to differentiate in the sense of distributions. But it doesn't make sense to differentiate in the usual sense, because $u$ is not necessarily differentiable. $\endgroup$
    – Ben McKay
    Mar 17, 2021 at 15:28
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    $\begingroup$ This is fairly standard PDE textbook stuff; try the textbooks of Folland or Rauch. $\endgroup$
    – Ben McKay
    Mar 17, 2021 at 15:29
  • $\begingroup$ @BenMcKay Thank you for the suggested books! $\endgroup$
    – ldgo
    Mar 17, 2021 at 15:41
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    $\begingroup$ note that any nonnegative distribution defines a (Radon) measure (essentially because positive linear functionals on the space of continuous functions are continuous, so a distribution that acts on test functions may not be continuous if we give the test function space the continuous functions topology but if the distribution is positive, it is continuous etc ) $\endgroup$
    – Conrad
    Mar 17, 2021 at 15:48

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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.

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  • $\begingroup$ Thank you! That is the relevant background that I am looking for. The reason why I decided to read an advanced book is that I hope I can find references along the way, now that I have got some references, I will read them first. Thank you again. $\endgroup$
    – ldgo
    Mar 18, 2021 at 1:50

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