# The composition of a L^2 function and a diffeomorphism is well-defined?

by perturbation   Last Updated January 11, 2019 12:20 PM - source

Let $$f \in L^2((0,1);\mathbb{R})$$ with respect to the Lebesgue measure. Let $$g$$ be a $$C^1$$-diffeomorphism from $$(0,2)$$ into its image $$g((0,2)) \subset (0,1)$$. Can I define the composition function $$h:(0,2) \rightarrow \mathbb{R}$$ by $$h(x)=f(g(x)),$$ for almost every $$x \in (0,2)$$ ? Is it enough to say that the measure of $$g((0,2))$$ is not zero ?

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