I need your help please, this is my problem
Let $(a,b)\subset \mathbb{R}$ and $F:(a,b)\to \mathbb{R}$ a convex function. Show that if $[c,d]\subset(a,b)$ is an compact interval, then exists $\alpha, \beta \in \mathbb{R}$ (constants) and a measure $\mu$ (which may depend on the interval [c,d]) such that $\forall x \in [c,d]$ you have the representation $$F(x)=\alpha x +\beta +\frac{1}{2}\int|x-y|\mu(dy)$$
My attempt:
$F$ is absolutely continuous in every compact $[c,d]\subset (a,b)$, then $F'$ is integrable and $$F(x)=F(c)+\int_c^xF'(u)du$$
I tried to get the representation but I failed. I had that $F'$ is a non decreasing function and it is right-continuous, then I can define $\mu_{F'}$ the Lebesgue-Stieltjes measure associate to $F'$ and I suspect that this is the $\mu$. If I take the integration by parts formula I get
$$F(x)=F(c)-cF(c)+xF(x)-\int y \mu_{F'}(dy)$$
And I don't know what to do, please help me.