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Math Help - Show that f o g is integrable on a bounded set A.

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by anlys View Post
    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}.
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