Let $\alpha$ be a monotonically increasing function on $(0,\infty)$. Let $g:(0,\infty)\rightarrow \mathbb{R}$ be a function such that $g\in\mathscr{R}(\alpha)$ on any bounded closed connected set. (That is, $[a,b]$)
Let $a>0$ be a real.
What i have learned is;
$\int_0^a g \, d\alpha=\lim_{t\to 0} \int_t^a g \, d\alpha$
And
$\int_a^\infty g \, d\alpha=\lim_{t\to\infty} \int_a^t g \, d\alpha$
And for $b>0$, $\int_a^b g \, d\alpha=\lim_{t\to b}\int_a^t g \, d\alpha$ if $\alpha$ is continuous at $b$.
(If these limits exist)
Here, what is the definition of $\int_0^\infty g \, d\alpha$ ?
Which limit should i take first? And what constraint gurantees that order of taking limits is irrelevant?