I am self-studying signed measure.I got stuck on the following remark:
Remark$\quad$ Let $M(X,\mathscr{A},\mathbb{R})$ be the collection of all finite signed measures on $(X,\mathscr{A})$. Then the formula $$ \mu\mapsto\int fd\mu $$ defines a linear functional on $M(X,\mathscr{A},\mathbb{R})$ if $f$ is an $\mathscr{A}$-measurable characteristic function.
So I can see that if $A\in\mathscr{A}$, then $\int\chi_Ad\mu=\mu(A)$ holds for each finite signed measure $\mu$. But why does that formula define a linear functional on $M(X,\mathscr{A},\mathbb{R})$? More specifically, why is the following true:
Let $(X,\mathscr{A})$ be a measurable space, let $A\in\mathscr{A}$, and let $f=\chi_A$ be the characteristic function of $A$. Define $F:M(X,\mathscr{A},\mathbb{R})\to\mathbb{R}$ by letting $$ F(\mu) = \int fd\mu. $$ Then $F(\mu+\nu)=F(\mu)+F(\nu)$ and $F(\alpha\mu)=\alpha F(\mu)$ holds for all $\alpha\in\mathbb{R}$ and all $\mu,\nu\in M(X,\mathscr{A},\mathbb{R})$.
I couldn't get why $\int fd(\mu+\nu) = (\mu+\nu)(A) = \mu(A)+\nu(A)$. Are we allowed to just write the last equality like that (for signed measure)?
Could someone please help me out? Thanks a lot in advance!
