$\int f \ d m = \int f c \ d \frac{m}{c}?$ Is it true that if $u = m/c$ is a measure, where $m$ is another measure and $c$ is a positive constant, then $\int cf \ du = \int f \ dm$? If not, what assumptions do we need for it to be true?
My attempt is to first note that it holds for simple functions, and then say that if $f$ is positive and measurable then letting $f_n$be simple and increasing simple functions: $$\int cf d u = \lim_{n \rightarrow \infty} \int cf_n d u = \lim_{n \rightarrow \infty} \int f_n d m = \int f \ d m$$
Is this reasoning correct? 

Yes. Since $u(E) = \frac 1c m(E)$ for every measurable set $E$ you have $\int \chi_E du = \frac 1c \int \chi_E dm$. Linearity extends this to simple functions, and monotone limits to arbitrary nonnegative measurable functions.
