Any positive distribution is a measure.

Question

Is it true that the limit of positive distributions is still a positive distribution?

I ask this, because I read that every positive distribution is a measure. So I would like to prove that if $(T_n)$ sequence of positive distributions such that $T_n\to T$, then $T$ is a positive distribution. Therefore by Riesz's Theorem, $T$ is a measure.

As a second question. Assuming that $T$ is a positive distribution. By Riesz's Theorem, exists $\mu$ a measure such that $T(\varphi)=\int_{\Omega}\varphi d\mu,\quad \varphi \in \mathcal{D}(\Omega):=\mathcal{C}_{0}^{\infty}(\Omega)$ with $\Omega$ open set of $\mathbb{R}^n.$ (test functions) But, why $T$ is a measure? By example, a requirement for something to be a measure is that the empty set takes it to zero. In our case, identifying $T$ with $T(\varphi)$ I imagine that we must have $T(\varphi(\emptyset))=\int_{\Omega\cap \emptyset}\varphi d\mu=0.$ but this does not seem accurate to me ...

Answer 1

The first question is immediate, since $\langle T,\phi\rangle=\lim \langle T_n,\phi\rangle$ is $\geq 0$ if each $\langle T_n,\phi\rangle$ is.

For what concerns the second question, we say that a distribution is a measure iff it acts on test functions by integration wrt a measure. So, it's just a definition.