Question regarding measure theory and Lebesgue integral

Let’s say we are given an integral $$int_{-infty}^{infty} f(x)g(x)dx$$ (Notation is Riemannian but we can see this as Lebesgue integral.)

Sometimes it’s convenient for us to see this as $$int_{-infty}^{infty} f(x)dmu (x)$$, where $$dmu (x)=g(x)dx$$

First of all, $dmu (x)$ and $dx$ are just notations, they aren’t numbers, so it seems to me that there has to be some way to justify this calculus.

Secondly, I’ve stumbled upon an explanation of this that says:
$gin L^1(-infty,infty)$, so we can define measure $mu$ in the following way:
$$mu (E)=int_E g(x)dx$$.

Although this does provide some clarification, I still don’t see why is just the condition $gin L^1(-infty,infty)$ relevant. Isn’t it necessary for $g$ to be nonnegative? Also, if we define $mu$ this way, I still think the identity
$$int_{-infty}^{infty} f(x)g(x)dx=int_{-infty}^{infty} f(x)dmu (x)$$ should be justified.

Is there a theorem that states this? Is it enough to prove this identity for simple (step) functions $f$ (which should be easy)?