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$A,B$ are blocks in $ \mathbb{R}^n$, and $f: A \rightarrow B$ a continuous map, such that $ ||f(x)-f(y)|| > c||x-y||$ for all $x,y \in A $ and some $c>0$. If $g: B \rightarrow \mathbb{R}$ is integrable. Prove that $g\circ f:A \rightarrow \mathbb{R}$ is integrable.

My approach is since $f$ is continuous then the set $D_f$ where $f$ is not continuous is empty so $D_f$ has null measure and $D_g$ is integrable then $D_g$ has also null measure. As $D_{g\circ f}\subset D_g\cup D_f$ then $D_{g\circ f}$ has null measure. By Lebesgue's Theorem $g\circ f$ is integrable.

Kutz
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  • You are mixing up Lebesgue and Riemann integrals. It is not true that continuous functions are Lebesgue integrable (when the domain is not known to be bounded) so your argument is not valid. – Kavi Rama Murthy Feb 24 '19 at 00:01
  • it's missing one more hypothesis that A and B are blocks in $\mathbb{R^n}$ – Kutz Feb 24 '19 at 00:08
  • But the question is about Lebesgue integration. Since the functions are not given to be bounded you cannot use continuity to get integrability. In particular, it is not given that $g$ is a bounded function, – Kavi Rama Murthy Feb 24 '19 at 00:12
  • The hyphotesis above $g$ is that $g$ is integrable. By definition $g$ has to be bounded. – Kutz Feb 24 '19 at 00:14
  • I've found the same question: https://math.stackexchange.com/questions/2463714/fx-fy-ge-cx-y-with-c0-then-for-gb-to-mathbbr-integrable-the?rq=1 – Kutz Feb 24 '19 at 00:42

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