Multivariable Advanced Calculus

296 INTEGRATION OF DIFFERENTIAL FORMSProposition 11.1.7 Let Ω be a P C 1 n dimensional manifold with boundary. Thenit is an orientable manifold if and only if there exists an atlas {(R i , U i )} such that foreach i, jdet D ( R i ◦ R −1 )j (u) ≥ 0 a.e. u ∈ int (Rj (U j ∩ U i )) (11.6)If v = R i ◦ R −1j (u) , I will often writeThus in this situation ∂(v 1···v n )∂(u 1···u n) ≥ 0.∂ (v 1 · · · v n )∂ (u 1 · · · u n ) ≡ det DR i ◦ R −1j (u)Proof: Suppose first the chart is an oriented chart sod ( v,A, R i ◦ R −1 )j = 1whenever v ∈ int R j (A) where A is an open ball contained in R i ◦ R −1j (U i ∩ U j \ L) .Then by Theorem 10.7.5, if E ⊆ A is any Borel measurable set,∫0 ≤R i ◦R −1j (A)X Ri◦R −1j (E) (v) 1dv = ∫Adet ( D ( R i ◦ R −1 ) )j (u) XE (u) duSince this is true for arbitrary E ⊆ A, it follows det ( D ( R i ◦ R −1 ) )j (u) ≥ 0 a.e. u ∈ Abecause if not so, then you could take E δ ≡ { u : det ( D ( R i ◦ R −1 ) ) }j (u) < −δ andfor some δ > 0 this would have positive measure. Then the right side of the above isnegative while the left is nonnegative. By the Vitali covering theorem Corollary 9.7.6,and the assumptions of P C 1 , there exists a sequence of disjoint open balls contained inR i ◦ R −1j (U i ∩ U j \ L) , {A k } such thatR i◦R −1j (A)int ( R i ◦ R −1j (U j ∩ U i ) ) = L ∪ ∪ ∞ k=1A kand from the above, there exist sets of measure zero N k ⊆ A k such that det D ( R i ◦ R −1 )j (u) ≥0 for all u ∈ A k \ N k . Then det D ( R i ◦ R −1 ) (j (u) ≥ 0 on int Ri ◦ R −1j (U j ∩ U i ) ) \(L ∪ ∪ ∞ k=1 N k) . This proves one direction. Now consider the other direction.Suppose the condition det D ( R i ◦ R −1 )j (u) ≥ 0 a.e. Then by Theorem 10.7.5∫d ( ∫v, A, R i ◦ R −1 )j dv = det ( D ( R i ◦ R −1 ) )j (u) du ≥ 0AThe degree is constant on the connected open set R i ◦ R −1j (A) . By Proposition 10.6.2,the degree equals either −1 or 1. The above inequality shows it can’t equal −1 and soit must equal 1. This proves the proposition. This shows it would be fine to simply use 11.6 as the definition of orientable in thecase of a P C 1 manifold and not bother with the definiton in terms of the degree. Thisis exactly what will be done in what follows. The version defined in terms of the degreeis more general because it does not depend on any differentiability.11.2 Some Important Measure Theory11.2.1 Eggoroff’s TheoremEggoroff’s theorem says that if a sequence converges pointwise, then it almost convergesuniformly in a certain sense.