Positive part of a distribution

Question

Call a distribution $u\in\mathcal{D}'(\mathbb{R})$ nonnegative if for all nonnegative test functions $\varphi$, $u(\varphi)\geq0$. What is known about when do distributions have positive and negative parts, i.e. when can one find for a given $u\in\mathcal{D}'(\mathbb{R})$, nonnegative distributions $u^+,u^-\in\mathcal{D}'(\mathbb{R})$ such that $u=u^+-u^-$?

Answer 1

This book says that a distribution is nonnegative if and only if it is given by a nonnegative measure. This means that the distribution ($u$ as well as $u^+$ and $u^-$) has to be of order zero. Thus, if $u = u^+ - u^-$ with $u^\pm \geq 0$ then $u$ must be the difference between two nonnegative measures and is therefore a signed measure. And since a locally finite signed measure gives a distribution, and a signed measure can be split into a positive and a negative part (by the Hahn Decomposition Theorem), then we have an "if and only if":

A distribution $u$ can be decomposed as $u = u^+ - u^-$ with $u^\pm \geq 0$ if and only if $u(f) = \int f(x) \, d\mu(x)$ for some locally finite measure $\mu$.