Multivariable Advanced Calculus

426 HAUSDORFF MEASURES AND AREA FORMULALemma 15.2.1 Let f : R n−1 → [0, ∞) be Borel measurable and letThen S is a Borel set in R n .S = {(x,y) :|y| < f(x)}.Proof: Set s k be an increasing sequence of Borel measurable functions convergingpointwise to f.Lets k (x) =∑N km=1c k mX E km(x).S k = ∪ N km=1 Ek m × (−c k m, c k m).Then (x,y) ∈ S k if and only if f(x) > 0 and |y| < s k (x) ≤ f(x). It follows thatS k ⊆ S k+1 andS = ∪ ∞ k=1S k .But each S k is a Borel set and so S is also a Borel set. This proves the lemma. Let P i be the projection ontospan (e 1 , · · ·, e i−1 , e i+1 , · · · , e n )where the e k are the standard basis vectors in R n , e k being the vector having a 1 in thek th slot and a 0 elsewhere. Thus P i x ≡ ∑ j≠i x je j . Also letA Pix ≡ {x i : (x 1 , · · · , x i , · · · , x n ) ∈ A}A Pi xxP i x ∈ span{e 1 , · · ·, e i−1 e i+1 , · · ·, e n }.Lemma 15.2.2 Let A ⊆ R n be a Borel set. Then P i x → m(A Pi x) is a Borelmeasurable function defined on P i (R n ).Proof: Let K be the π system consisting of sets of the form ∏ nj=1 A j where A i isBorel. Also let G denote those Borel sets of R n such that if A ∈ G thenP i x → m((A ∩ R k ) Pix) is Borel measurable.where R k = (−k, k) n . Thus K ∈ G. If A ∈ G( (AP i x → mC ) )∩ R k P i xis Borel measurable because it is of the formm ( (R k ) Pi x)− m((A ∩ Rk ) Pi xand these are Borel measurable functions of P i x. Also, if {A i } is a disjoint sequence ofsets in G thenm ( ) ∑(∪ i A i ∩ R k ) Pi x = m ( )(A i ∩ R k ) Pi xi)