Let $a,b\in\mathbb{R}, a < b, f,g\in L_1([a,b])$ and for $x\in[a,b]$ let $F(x):=\int_{(a,x)}f d\mathcal{L}^1,\; G(x):=\int_{(a,x)}gd\mathcal{L}^1$.
Show that $F$ and $G$ are continuous and that $\int_{(a,b)}Fgd\mathcal{L}^1=F(b)G(b)-\int_{(a,b)}fGd\mathcal{L}^1$.
I've been able to show the continuity of $F$ and $G$ using measure theoretic induction. From that it follows that $\int_{(a,b)}Fgd\mathcal{L}^1$ and $\int_{(a,b)}fGd\mathcal{L}^1$ exist, since $F$ and $G$ are bounded on $[a,b]$. My problem is: How do I show the last equality?