Integrals of Subharmomc FuncUons on Manifolds of NonnegaUve Curvature 279

With g{p)ETMp for aWpeU and with

^ ( { pGL / |g ( p ) Ф ( gradcp ) ( p ) } ) <^ / 2

Lusin's theorem holds here (applied to sections of a vector bundle) cause it IS essentially a local statement the usual proof can be applied to any open set over which the bundle is trivial and so the required globally defined function can be obtained by a standard partion-of-unity construction

hti E, = {peU\g{p) = {gv^d(p)(pl\\g(p)\\^^}^ Then since ||gradç>|| ^ 1 almost everywhere, pi{U-E^)<r]/2 Now let £2 be the set of points of £1 at which £1 has Lebesgue density 1 By the Lebesgue density theorem ([21, p 129]),/i(£,-E2) = 0 UQncQ^(U-E2) = fi(U-Ei)-{-ß{E,-E2)< г]/2 To complete the proof of the lemma, it suffices to show that for each peE2 there exists a 5p>0 such that for all ье(0, (5Д ||(grad (p,){p)\\ >l-rj

For in that case if (7, = {pe[/|||(grad c^J(p)|| > 1->7 for all f g(0, 1/0},

+ 00

f = 1,2,3, then C/i с (72 с [/3 с and IJ ^« =" ^2 Thus there is an Iq

1=1 such that 1л{Е2-и^<г]/2 so that ^u({/7eK|||grad (p,\\ < \-rj}^fi{U-E2) + fi{E2-UJ<rf/2 + rj/2 = ri for all гб(0, lAo)

To show that for each peE2, ||(grad</)J(p)|| > 1-^ for ье{0^р) if ^p>0 IS sufficiently small, first suppose that ô^ is chosen so small that

Qpi { vETMp\QxppVeE2A\v\\<op } ) > { l - y ) Op ( { veTMp\\\v\\<ôp } )

where as before Qp is the measure on TMp obtamed from the inner product on TMp determined by the Riemanman metric on M and where

7 IS a positive number less than

8 being a Lipschitz constant for cp on U and к„ bemg the volume of the unit ball m R" Note that if G is a measurable subset of the unit ball Б(1) in R" whose Lebesgue measure /x(G) is greater than 1 -7 for such a 7 then

lK { \\u\\ ) dfi^ j /c(||t;||)^^-(max/c)^(5(l)-G) >1-(тахм:)(7)>-

' - T

That such a choice oïôp is possible follows from the facts that the Lebesgue density of £2 at p IS 1 and that exp^ has continuous Jacobian determinant