Let $f:[0,+\infty)\to[0,+\infty)$ and $g:[0,+\infty)\to(0,+\infty)$ be two Lebesgue integrable functions such that $$\lambda(t)=\frac{f(t)}{g(t)}:[0,+\infty)\to[0,+\infty)$$
is non-increasing.
Show that $$F(t)=\frac{\int_0^t f(x)\mathrm{d}x}{\int_0^t g(x)\mathrm{d}x}$$ is again non-increasing. Furthermore, if $F(t_1)=F(t_2)$ for $t_1<t_2$, then $\lambda$ is (a.e.) a constant on $[0,t_2].$
My attempts
Life will be a lot easier if $f,g$ are continuous functions. In that case, $F$ is differentiable and everything follows easy calculation: $$\frac{\mathrm{d}}{\mathrm{d}t}F(t)=\frac{f(t)\int_0^t g-g(t)\int_0^{t}f}{\left(\int g\right)^2}=\frac{g(t)}{(\int g)^2}\int_0^t g(x)\left(\frac{f(t)}{g(t)}-\frac{f(x)}{g(x)}\right)\mathrm{d}x=\frac{g(t)}{(\int g)^2}\int_0^t g(x)(\lambda(t)-\lambda(x))\mathrm{d}x\leq 0$$
How to deal with integrable functions? I tried some approximation to $f,g$ using continuous functions but got lost. Thanks!