Integration with respect to a signed measure Let $\mu$ be a singed measure, $f\in C_c(X)$, I want to show
$$\int fd(c\mu) = c \int fd\mu, \forall c \in \mathbb{R}$$
Since $c\mu$ is also a singed measure, I think by definitionm I need to show  $$\int fd(c\mu)^+-\int fd(c\mu)^- = c \int fd\mu^+-c \int fd\mu^-$$
But I think here I can't directly say $d(c\mu)^+=cd\mu^+,d(c\mu)^-=cd\mu^-$, since the definition of $\mu^+,\mu^-$ involves $|\mu|$ and $|c\mu|=|c||\mu|$, hence I need to discuss wether $c<0$ or $c>0$ or $c=0$.
For example, $c<0$, then $$(c\mu)^+=\frac{1}{2}(|c\mu|+c\mu)=-c\mu^-$$$$(c\mu)^-=\frac{1}{2}(|c\mu|-c\mu)=-c\mu^+$$
Hence the LHS equals to the RHS since $\int fd(c\lambda) = c \int fd\lambda, \forall c >0$ when $\lambda$ is a measure.
Since I am new to this subject, could you help to confirm whether my understanding is correct?

Seems correct, I assume that you used the following line of reasoning: if $c<0$ then
$$
  (c\mu)^+ = \frac12(|c\mu|+c\mu) = \frac12|c|(|\mu|-\mu) = |c|\mu^- = -c\mu^-.
$$
Alternatively, you could go via the decomposition of the state space into positive and negative sets.
