Consider $T>0,$ a continuous function $f:[0,T] \to \mathbb{R}.$
Prove that for $0 \leq u \leq T,\lim_{\beta \to \infty}\frac{1}{\beta}\ln\left(\int_0^u e^{\beta f(r)}dr\right)=\sup_{r \in [0,u]}f(r)$ and that the convergence is uniform over the set of functions having the same modulus of continuity.
The first question follows from the fact that $\lim_{p \to \infty}||g||_p=||g||_{\infty}$ applied on $([0,u],\mathcal{B}([0,u]),\lambda)$ for $g=e^f.$
How to prove the second part, that is, denoting $\mathcal{F}$ the set of functions with the same modulus of continuity $\omega,\lim_{\beta \to \infty}\sup_{f \in \mathcal{F}}|\frac{1}{\beta} \ln \left(\int_0^ue^{\beta f(r)}dr\right)-\sup_{r \in [0,u]}f(r)|=0?$