Proposition 2.6.14DFD880D9E4D

Proof. For a nonnegative test function $\chi $, the numbers $\langle T_ i,\chi \rangle $ are nonnegative, hence so is their limit $\langle T,\chi \rangle $; likewise $\langle T,\chi \rangle $ is real for real $\chi $. Thus $T$ is positive, and $T$ and the $T_ i$ are given by Radon measures $\mu $ and $\mu _ i$. Let $f\in C_ c(\Omega )$. By lemma 2.5.1(5), $f$ is the uniform limit of test functions supported in a fixed compact set $L$; in particular, $f$ vanishes outside $L$. If $\eta $ is a smooth cutoff function for $L$ in $\Omega $, then $\mu _ i(L)\leq \langle T_ i,\eta \rangle \to \langle T,\eta \rangle $. A convergent net of real numbers need not be bounded, but it is bounded from some index on; hence there is $i_0$ with $\mu _ i(L)\leq C=\langle T,\eta \rangle +1$ for $i\geq i_0$, and also $\mu (L)\leq \langle T,\eta \rangle <C$. For every test function $\chi $ supported in $L$ and every $i\geq i_0$,

so $\varlimsup _ i|\int f\, \mathrm d\mu _ i-\int f\, \mathrm d\mu |\leq 2C\| f-\chi \| _\infty $, which can be made arbitrarily small. Hence $\mu _ i\to \mu $ vaguely. The converse holds because test functions belong to $C_ c(\Omega )$. $\square$