Corollary 2.8.2D30ED2659FC7

Proof. (1) Denote also by $C\nu $ the positive distribution defined by integration against the Radon measure $C\nu $. For every nonnegative test function $\chi $ we have $\langle C\nu \pm T,\chi \rangle =C\int \chi \, \mathrm d\nu \pm \langle T,\chi \rangle \geq 0$, so the real distributions $C\nu \pm T$ are positive, and by theorem 2.5.6 they are given by Radon measures $\lambda _+$ and $\lambda _-$. The Radon measures $\lambda _++\lambda _-$ and $2C\nu $ define the same distribution, so they are equal by proposition 2.5.3(4). In particular, $\lambda _+(E)\leq 2C\nu (E)$ for every Borel set $E\subset \Omega $, and lemma 2.8.1(1) gives $\lambda _+=h\nu $ for a Borel function $h$ with values in $[0,2C]$. Hence $\langle T,\chi \rangle =\int \chi \, \mathrm d\lambda _+-C\int \chi \, \mathrm d\nu =\int \chi (h-C)\, \mathrm d\nu $ for every test function $\chi $, and $g=h-C$ has the required properties.

Uniqueness. Let $g$ and $g'$ be real Borel functions, locally integrable with respect to $\nu $, with $\int \chi g\, \mathrm d\nu =\int \chi g'\, \mathrm d\nu $ for every test function $\chi $, and put $w=g-g'$. The Borel measures $w^+\nu $ and $w^-\nu $, where $w^\pm =\max \{ \pm w,0\} $, are finite on compact sets, hence Radon by proposition 2.2.2, and they define the same distribution. By proposition 2.5.3(4) they are equal, and evaluating them on the set $\{ w>0\} $ gives $\int _{\{ w>0\} }w\, \mathrm d\nu =0$, so that $\nu (\{ w>0\} )=0$. Similarly $\nu (\{ w<0\} )=0$, and $g=g'$ $\nu $-almost everywhere.

(2) Let $\mathcal U$ be the family of the open sets $U$ in the hypothesis. Each set of a compact exhaustion of $\Omega $ is covered by finitely many members of $\mathcal U$, so countably many members $U_1,U_2,\ldots $ of $\mathcal U$ cover $\Omega $. Let $U\in \mathcal U$. The restriction of $\nu $ to the Borel subsets of $U$ is a Radon measure on $U$ by proposition 2.2.2, and test functions on $U$, extended by zero, are test functions on $\Omega $. Hence (1), applied on $U$, gives a Borel function $g_ U$ on $U$ with values in $[-C_ U,C_ U]$ which represents $T$ on $U$. By the uniqueness assertion, applied on the open sets $U_ k\cap U_ l$, we have $g_{U_ k}=g_{U_ l}$ $\nu $-almost everywhere on $U_ k\cap U_ l$. Define $g=g_{U_ k}$ on $P_ k=U_ k\setminus \bigcup _{l<k}U_ l$. Since $P_ l\cap U_ k=\varnothing $ for $l>k$, we have $g=g_{U_ k}$ $\nu $-almost everywhere on $U_ k$ for every $k$. Hence $g$ is locally integrable with respect to $\nu $, and $\langle T,\chi \rangle =\int \chi g\, \mathrm d\nu $ for every test function $\chi $ supported in some $U_ k$; by lemma 2.5.2, this holds for every test function. Finally, for every $U\in \mathcal U$, the uniqueness assertion on $U$ gives $g=g_ U$ $\nu $-almost everywhere on $U$, so $|g|\leq C_ U$ $\nu $-almost everywhere on $U$. $\square$