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Let $g$ be a continuous function defined on the set $\mathcal{I}\times\mathcal{X}$. If $\mathcal{I}$ and $\mathcal{X}$ are compact sets, then how to prove the followings:

  1. $g$ is uniformly continuous

  2. If $E$ is a measurable subset of $\mathcal{I}$ with measure, satisfying $\left|E\right|<\delta$, where $\left|.\right|$ indicates the measure, then for a given integrable function $\mu\left(t\right)$ defined on $\mathcal{I}$, following is true \begin{equation} \int_E \mu\left(t\right)dt < \epsilon \end{equation}

Source: These equations are taken from the proof of Chattering Lemma , page 132-33, Optimal Control Theory, by L. Berkovitz. Since I dont have much understanding of ananlysis, I am finding it difficult to go through these proofs. Any help would be appreciated.

user146290
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    For question one, you can see http://math.stackexchange.com/questions/110573/continuous-function-on-a-compact-metric-space-is-uniformly-continuous –  Nov 30 '14 at 04:48
  • It is unclear what $\epsilon$ and $\mu$ is in 2. –  Nov 30 '14 at 04:49
  • $\mu$ is any integrable function defined on the set $\mathcal{I}$. Another definition of $\mu$ given in the proof is the following. If $f$ is a measurable function on $\mathcal{I}$ for each $x\in\mathcal{X}$ and is continuous on $\mathcal{X}$ for each $t\in \mathcal{I}$, then for all $\left(t,x\right)$ and $\left(t,x'\right)$ on $\mathcal{I}\times\mathcal{X}$, $\left|f\left(t,x\right)-f\left(t,x'\right)\right|<\mu(t)\left|x-x'\right|$. – user146290 Nov 30 '14 at 04:59
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    What you need for question 2 appears to be the notion of uniform integrability. Are you familiar with that? Once you prove that an integrable function is uniformly integrable, applying the definition will do. – Franco Nov 30 '14 at 06:37

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