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I'm reading the proof of change of variable in Spivak's Calculus on manifolds.

In the last inequality of page 68, I think this proof assumed that $ f\circ g |\det g'| $ is integrable on $g^{-1}(V)$ . But Cannot understand why this is true. Can anyone explain why this function is integrable?

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In the second reduction, we have assume that the theorem is true for a constant function. $f_S$ is such a function. So using our hypothesis we obtain $$\int_{intS}f_s = \int_{g^{-1}(intS)}(f_s\circ g |\det g'|).$$

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