ABSTRACT
P roof. Given an α-division K, of K let /Co consist of all members of /C at which 5 = 0, and K\ consist of all at which 5 > 0. By hypothesis all members of /Ci are /3-fine. Let B be the figure partitioned by /Co· Take an arbitrary /3-division B of B and let K! be the /3-division K\ \JB of K . Then ( Σ 5)(/C) = (Σ ^ Χ Κ Ο < (E S )(K i) + = ( E W ) < sW (K ). That is, ( Σ S)()C) < S ^ \ K ) for every α-division K of K. Hence, S^a\ K ) < S ^ \ K ) . □
Lemma B. Let the summants Tj > 0 on K for i — 0,1,2, · · · such that given 0 < c < 1 there exists an integer-valued func tion n(t) > 1 on K satisfying
(1)
for every tagged cell (I, t) in K . Then
P roof. Define the summant Si on K for i = 1,2,··· by letting S i(I ,t) = T{(I,t) for i < n (t) ,0 for i > n(t). Then cTq < Σ Ζ ι s i by (1)· So for every gauge a on K
Let ε > 0 be given. Since 0 < Si < T{ for all i > 1 we can choose for each such i a gauge β{ on K small enough so that
for i = 1 ,2 ,···. Let a(t) be the minimum value of β{(ί) for i = 1, · · · , n(t). Then by Lemma A
since a(t) < /?j(t) if S i(I ,t) > 0. By (3), (5) and (4) c fKTo < Σ Ζ ι Ι κ Τί + ε · Letting ε —► 0+ and then c —> 1-we get (2). □
I f moreover JK ViS exists for i = 1 , 2,··· and v0 — yi then
(7)
(8)
Given 0 < c < 1 consider any point t in K . If v0(t) > 0 then cv0(t) < va(t) < Y ^ iV i ( t ) . So there is a smallest positive integer n(t) such that
If on the other hand v0{t) = 0 then (9) holds for n(t) = 1. For any cell I with endpoint t, multiplying (9) through by S ( I , t ) gives (1) in Lemma B under (8). So Lemma B gives (6) by (2) and (8).
