Let $F$ denote a distribution on $C_c^{\infty}$ or $S(R^d)$ , can we decomposite $F$ as the difference of two positive distribution $F^+-F^-$ ?
Here positive distribution means whenever $\varphi(x) \ge 0$ for all $x \in R^d$ , then we have $F(\varphi)\ge 0$ .
First , note that if $$F(\varphi)=\int f(x)\varphi(x)\, dx$$ Let $E$ denote the set which $f(x)\ge 0$ , then we have $$F(\varphi)=\int f(x)\chi_E(x)\varphi(x) \,dx + \int f(x) \chi_{E^c}(x) \varphi(x) \,dx$$
Then recall that if $F$ is a distribution , we can find a sequence of function $f_n \in C^{\infty}$ such that $$F(\varphi)=\lim_{n\to \infty}\int f_n(x) \varphi(x) \, dx$$ We can define $$F^+(\varphi)=\lim_{n\to \infty}\int f_n(x)\chi_{E_n}(x) \varphi(x)\, dx$$ and $$F^-(\varphi)=-\lim_{n\to \infty}\int f_n(x)\chi_{E_n^c}(x) \varphi(x)\, dx$$
Then we need to prove the following two proposition:
$(1)$ The limit as $n\to \infty $ exist for $F^+$ and $F^-$ .
$(2)$ Whenever $\varphi_n\to \varphi$ , then we have $F^+(\varphi_n)\to F(\varphi)$ .
The problem is to deal with $f_n$ . I want to set $\{\phi_n\}$ be an approximation to the identity supported in a compact set and $f_n(x)=F(\phi_n(x-y))$ but I don not know how to deal with this next .