Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
What does "(another question)" in the title mean? –  joriki May 4 '13 at 20:19

1 Answer 1

Use the monotone convergence theorem or the Dominated Convergence Theorem. For a more general aproch use the following result:

Theorem. Let $\{ F_t ; t\in T\}$ be a family of functions $F_t : X > \rightarrow \mathbb{C}$ depending on a parameter t; let $\mathcal{B}_X$ be a topological base $X$ and $\mathcal{B}_{T}$ a topological base in $T$. If the family converges uniformly on $X$ over the base $\mathcal{B}_{T}$ to a function $F : X \rightarrow \mathbb{C}$ and the limit $\lim_{\mathcal{B}_{T}} F_t(x)=A_t$ exists for each $t\in T$, the both repeated limits $\lim_{\mathcal{B}_{X}}(\lim_{\mathcal{B}_{T}}F_t(x))$ and $\lim_{\mathcal{B}_{T}}(\lim_{\mathcal{B}_{X}}F_t(x))$ exist and the equality

Proof. See Zoric. P. 381.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.