U onto r -1 conf'ormally. and in k ,span[z 'l'hen qJ :k I -11 ) > o. extends t o a omeomorphism h of c( 2Jt::,) -1 k dist(cp ,span[q:>; k f -l}) > o. t By Mergelyan '8 theorem the coordinate generated by ~ on 66. Hence the F. and M. Izl=r I(z h(Z) ~ qJ'(z)/21r1. for h with k > 0 and by HI. We can € 0 < r < I, and ~(z)h(z)dz = 0 • )
Proof: By definition, formula. ns rl · , 1/ 1 Jv 7r for all g € C~. 1( z ) dxdy "0-; Since by Green's theorem o ~ g(O 1 ~dxdy (j-; '" we get (1. 3) via. Fubini's theorem. 3: If ~ on an open set V, then Proof: is almost everywhere equal to a function analytic I~I (V) = O. 1. , then C: ~ ~ o. in the compactly supported continuous functions. 5) and the obviously necessary conditions actually determine the Cauchy transform ~ of ~. ",f(z) = 0 allu let iJ. be a. -39- compactly supported measure such that fez} Then Proof: ~ (If iJ-; 0 almost everywhere.
8: ~ E C(E,l). Let §3. I~I(L); 0 Prove that I~I (J) ; 0 tha,t be a measure on a compa,ct set ~ J if E such that for every straight line L. Prove is a rectifiable curve. riation, and s. Theorem of Havin Throughout this sect ion we fix a. compact set f, supposed, as before, to be analytic on We want to know when there is a meas ure z (E. s a measurable extension at bounded var i ation on C. f It is easier t o attack the problem directly. inleve length if there is a number U ~ consists of finitely many analytiC Jordan curves surrounding in the usual sense of contour integration.
Analytic Capacity and Measure by J. Garnett