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Suppose that $\varphi$ is a smooth $\mathcal{K}_{\infty}$-function and $\bar{\textbf{B}}$ is the unit ball set in $\mathbb{R}^{n}$. Let $x : \mathbb{R}_{\geq 0} \times \mathbb{R}^{n} \to \mathbb{R}^{n}$ be such that $x(.,\xi)$ is differentiable for each $\xi$ and $x(t,.)$ is uniformly locally Lipschitz for each $t$. Also let $\omega : \mathbb{R}^{n} \to \mathbb{R}_{\geq 0}$ be a continuous function on $\mathbb{R}^{n}$ with $\omega(0) = 0$ and $\omega(x) \to \infty$ as $x \to \infty$. I'd like to know that whether the following holds $$ \sup_{\xi \in \bar{\textbf{B}}} \int_{s=0}^{s=t}{\varphi(\omega(x(s,\xi)))ds} = \int_{s=0}^{s=t}{\sup_{\xi \in \bar{\textbf{B}}} \varphi(\omega(x(s,\xi)))ds}. $$ In other words, are the supremum and integral signs interchangable?

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