I'm studying continuous martingales and there is this thing which worries me:
Let $a$ be of finite variation and $f$ be $a$–integrable. Then the function $(f · a)$ is right-continuous and of finite variation, where $(f · a)$ is defined as
$$\int_0^tf(s)\mu_+(ds)-\int_0^tf(s)\mu_-(ds)$$
where $\mu_+((0,t])=\frac{V_a(t)+a(t)}{2}$ and $\mu_-((0,t])=\frac{V_a(t)-a(t)}{2}$
where $V_a(t)$ is the variation of $a$ on $(0,t)$.
My lecture notes claim that assosiativity, i.e. $g·(f·a)=(gf)·a$ follows "readily", but I can't see it - the problem is that if I try to evaluate $g·(f·a)$ I get stuck with using $V_{f.a}$ bit in the measure.
Any help is appreciated!