Thread: Show that f o g is integrable on a bounded set A.

1. Show that f o g is integrable on a bounded set A.

Hello,
I need help with the following problem. Can anyone help me on this? Thank you in advance.

Problem:
Let A in Rn be bounded with volume.
Suppose that U is an open subset of Rn such that the closure of A cl(A) is contained in U , and that g:U-->Rn is one-to-one and g^-1 is C1 on g(U) with Jg-1(x) ≠ 0 for all x in g(A).
Prove that if f is integrable on g(A), then f o g is integrable on A.

2. Originally Posted by anlys
Hello,
I need help with the following problem. Can anyone help me on this? Thank you in advance.

Problem:
Let A in Rn be bounded with volume.
Suppose that U is an open subset of Rn such that the closure of A cl(A) is contained in U , and that g:U-->Rn is one-to-one and g^-1 is C1 on g(U) with Jg-1(x) ≠ 0 for all x in g(A).
Prove that if f is integrable on g(A), then f o g is integrable on A.
You want to show that $\int_{A} |f(g(x))|dx<\infty$. Just apply the change of variable formula (the hypotheses allow you to do so) and use the fact that $Jg^{-1}$ is bounded on $g(A)$ by continuity of $dg^{-1}$.