Let $F(t) = \int_{\mathbb{R}_{\ge 0}} e^{-tx^2}dx = \int_{\mathbb{R}_{\ge 0}} f$
I want to show that $F(t)$ exists and is continuous for $t > 0$ whereby $\underset{t \downarrow 0+}{lim}$ $F(t) = +\infty$
$\fbox{Idea to Prove the Existence of F(t)}$
One could use the Monotone Convergence Theorem to prove the existence of $F(t)$. Specifically, if we could find a sequence of measurable functions $\{f_n\}$ s.t. $0 \le f_n \le e^{-tx^2}$ for all $n$ and $f_n \rightarrow e^{-tx^2}$ pointwise on $\mathbb{R}_{\ge 0}$, then it would follow that $e^{-tx^2}$ is measurable with
$ lim \int_{\mathbb{R}_{\ge 0}} f_n = \int_{\mathbb{R}_{\ge 0}} e^{-tx^2} = F(t) $