Multivariable Advanced Calculus

9.8. CHANGE OF VARIABLES FOR LINEAR MAPS 233Corollary 9.8.4 Suppose A ∈ L (R p , R p ). Then if S is a Lebesgue measurable set,it follows AS is also a Lebesgue measurable set.In the next lemma, the norm used for defining balls will be the usual norm,( p∑) 1/2|x| = |x k | 2 .k=1Thus a unitary transformation preserves distances measured with respect to this norm.In particular, if R is unitary, (R ∗ R = RR ∗ = I) thenR (B (0, r)) = B (0, r) .Lemma 9.8.5 Let R be unitary and let V be a an open set. Then m p (RV ) =m p (V ) .Proof: First assume V is a bounded open set. By Corollary 9.7.6 there is a disjointsequence of closed balls, {B i } such that U = ∪ ∞ i=1 B i ∪ N where m p (N) = 0. Denote byx i the center of B i and let r i be the radius of B i . Then by Lemma 9.8.1 m p (RV ) =∑ ∞i=1 m p (RB i ) . Now by invariance of translation of Lebesgue measure, this equals∞∑∞∑m p (RB i − Rx i ) = m p (RB (0, r i )) .i=1Since R is unitary, it preserves all distances and so RB (0, r i ) = B (0, r i ) and therefore,i=1i=1∞∑∞∑m p (RV ) = m p (B (0, r i )) = m p (B i ) = m p (V ) .This proves the lemma in the case that V is bounded. Suppose now that V is just anopen set. Let V k = V ∩ B (0, k) . Then m p (RV k ) = m p (V k ) . Letting k → ∞, this yieldsthe desired conclusion. This proves the lemma in the case that V is open. Lemma 9.8.6 Let E be Lebesgue measurable set in R p and let R be unitary. Thenm p (RE) = m p (E) .Proof: Let K be the open sets. Thus K is a π system. Let G denote those Borelsets F such that for each n ∈ N,i=1m p (R (F ∩ (−n, n) p )) = m n (F ∩ (−n, n) p ) .Thus G contains K from Lemma 9.8.5. It is also routine to verify G is closed with respectto complements and countable disjoint unions. Therefore from the π systems lemma,G ⊇ σ (K) = B (R p ) ⊇ Gand this proves the lemma whenever E ∈ B (R p ). If E is only in F p , it follows fromTheorem 9.4.2E = F ∪ Nwhere m p (N) = 0 and F is a countable union of compact sets. Thus by Lemma 9.8.1m p (RE) = m p (RF ) + m p (RN) = m p (RF ) = m p (F ) = m p (E) .This proves the theorem.