I've come across situations where, say, we're given a locally integrable function $f$ and measure $\mu$, and for whatever reason we find ourselves working with the mapping $$g\to \int gf\,d\mu.$$ At times, it seems to be a sensible thing to suddenly "regard $f\,d\mu$ as a measure in its own right" (say $d\,\lambda=f\,d\mu$) and proceed to work with the mapping $$g\to \int g\,d\lambda.$$ What exactly is the point of this? I realize that the answer will probably depend on the context, but I intentionally want the question to be a little broad. To perhaps get to the crux of the question, I should say...there must be a reason measure theory is studied in its full generality instead of us just learning that there's something called a Lebesgue measure (and counting measure and a few others, sure). So why do we come across situations where we need to construct measures? Regard something as a measure? Etc. Etc.
I apologize if the question is terribly vague, but this also means I'll appreciate just about any well thought out answer.