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Let $g:D \subset \mathbb R^{1+n} \to \mathbb R^m$ be a continuous function and $\mu(t):\mathbb R \to \mathbb R^n$ also a continuous function. If I chose a compact $K \subset D$ then I can define $M := \max\{||g(t,x)|| ~|~ (t,x) \in K \}$. My question is if I can make the following estimate and if yes is there a common name for this estimate. \begin{equation*} \int_a^bg(t,\mu(t))dt \leq \int_a^bM dt = M|b-a| \end{equation*}

Does this work? Thanks!

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    You need $(t,\mu(t)) \in K$ for $t\in [a,b]$. If you have that, you can make this estimate. – 2017-01-24
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    @DanielFischer Ah yes I forgot this condition. Yes this condition is satisfied. Is there a name for this estimate? – 2017-01-24
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    It's sometimes called the standard estimate, $$\biggl\lvert \int_X f\,d\mu\biggr\rvert \leqslant \mu(X)\cdot\sup \{ \lvert f(x)\rvert : x \in X\}.$$ For path integrals (and integrals over a bounded real interval are path integrals), it's also known as the ML estimate or ML inequality (ML for maximum [of the (absolute value of the) function] and length [of the path]). – 2017-01-24

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