Multivariable Advanced Calculus

9.9. CHANGE OF VARIABLES FOR C 1 FUNCTIONS 239Theorem 9.9.4 Let U and V be open sets in R p and let h, h −1 be C 1 functionssuch that h (U) = V. Then if g is a nonnegative Lebesgue measurable function,∫∫g (y) dm p = g (h (x)) |det (Dh (x))| dm p . (9.13)VUProof: From Corollary 9.9.3, 9.13 holds for any nonnegative simple function in placeof g. In general, let {s k } be an increasing sequence of simple functions which convergesto g pointwise. Then from the monotone convergence theorem∫∫∫g (y) dm p = lim s k dm p = lim s k (h (x)) |det (Dh (x))| dm pVk→∞ Vk→∞ U∫= g (h (x)) |det (Dh (x))| dm p .UThis proves the theorem. Of course this theorem implies the following corollary by splitting up the functioninto the positive and negative parts of the real and imaginary parts.Corollary 9.9.5 Let U and V be open sets in R p and let h, h −1 be C 1 functionssuch that h (U) = V. Let g ∈ L 1 (V ) . Then∫∫g (y) dm p = g (h (x)) |det (Dh (x))| dm p .VUThis is a pretty good theorem but it isn’t too hard to generalize it. In particular, itis not necessary to assume h −1 is C 1 .Lemma 9.9.6 Suppose V is an p−1 dimensional subspace of R p and K is a compactsubset of V . Then lettingK ε ≡ ∪ x∈K B (x,ε) = K + B (0, ε) ,it follows thatm p (K ε ) ≤ 2 p ε (diam (K) + ε) p−1 .Proof: Using the Gram Schmidt procedure, there exists an orthonormal basis forV , {v 1 , · · · , v p−1 } and let{v 1 , · · · , v p−1 , v p }be an orthonormal basis for R p . Now define a linear transformation, Q by Qv i = e i .Thus QQ ∗ = Q ∗ Q = I and Q preserves all distances and is a unitary transformationbecause ∣ ∣∣∣∣Q ∑ ∣ ∣∣∣∣2 ∣ ∑ ∣∣∣∣2a i e i =a∣ i v i = ∑ ∣ |a i | 2 ∑ ∣∣∣∣2=a∣ i e i .iiiThus m p (K ε ) = m p (QK ε ). Letting k 0 ∈ K, it follows K ⊆ B (k 0 , diam (K)) and so,QK ⊆ B p−1 (Qk 0 , diam (QK)) = B p−1 (Qk 0 , diam (K))where B p−1 refers to the ball taken with respect to the usual norm in R p−1 . Everypoint of K ε is within ε of some point of K and so it follows that every point of QK ε iswithin ε of some point of QK. Therefore,QK ε ⊆ B p−1 (Qk 0 , diam (QK) + ε) × (−ε, ε) ,i