Multivariable Advanced Calculus

234 THE LEBESGUE INTEGRAL FOR FUNCTIONS OF P VARIABLESLemma 9.8.7 Let D ∈ L (R p , R p ) be of the formD = ∑ jd j e j e jwhere d j ≥ 0 and {e j } is the usual orthonormal basis of R p . Then for all E ∈ F pm p (DE) = |det (D)| m p (E) .Proof: Let K consist of open sets of the form{p∏p∑}(a k , b k ) ≡ x k e k such that x k ∈ (a k , b k )k=1k=1Hence)}( p∏k=1{ p∑k=1D(a k , b k )=d k x k e k such that x k ∈ (a k , b k )=p∏(d k a k , d k b k ) .k=1It follows( p∏))m p(D (a k , b k )k=1( p∏) ( p∏)= d k (b k − a k )k=1 k=1( p∏)= |det (D)| m p (a k , b k ) .k=1Now let G consist of Borel sets F with the property thatm p (D (F ∩ (−n, n) p )) = |det (D)| m p (F ∩ (−n, n) p ) .Thus K ⊆ G.Suppose now that F ∈ G and first assume D is one to one. Thenand som p(D(F C ∩ (−n, n) p)) + m p (D (F ∩ (−n, n) p )) = m p (D (−n, n) p )m p(D(F C ∩ (−n, n) p)) + |det (D)| m p (F ∩ (−n, n) p ) = |det (D)| m p ((−n, n) p )which shows( (m p D F C ∩ (−n, n) p)) = |det (D)| [m p ((−n, n) p ) − m p (F ∩ (−n, n) p )]=(|det (D)| m p F C ∩ (−n, n) p)In case D is not one to one, it follows some d j = 0 and so |det (D)| = 0 andso F C ∈ G.0 ≤ m p(D(F C ∩ (−n, n) p)) ≤ m p (D (−n, n) p ) == |det (D)| m p(F C ∩ (−n, n) p)p∏(d i p + d i p) = 0i=1