# Interchanging supremum and integral

Suppose that $\varphi$ is a smooth strictly increasing function with $\varphi(0)=0$ and $B$ is a compact subset of $R^{n}$. Let $x : R_{\geq 0} \times R^{n} \to R^{n}$ be such that $x(.,\xi)$ is differentiable for each $\xi$ and $x(t,.)$ is uniformly locally Lipschitz for each $t$ (i.e. there is some strictly positive constant $L$ such that $\| x(t,\xi_{1}) - x(t,\xi_{2})\| \leq L \| \xi_{1} - \xi_{2} \|$ for all $\xi_{1},\xi_{2} \in K \subset R^{n}$). I'd like to know that whether the following holds $$\sup_{\xi \in B} \int_{s=0}^{s=t}{\varphi(x(s,\xi))ds} = \int_{s=0}^{s=t}{\sup_{\xi \in B} \varphi(x(s,\xi))ds}.$$ In other word, are the supremum and integral signs interchangable?

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There is a type in the question. Please replace $R^{n}$ by $R$ –  Navid Noroozi Dec 4 '12 at 6:59

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