By J. Garnett

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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. open V ~ E contains regular neighborhood V £ of such that every E such that dV has length at most £.

1: Klz - wi. 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.

Bly additive, bounded extension This can be seen via the Riesz representation theorem, or directly using the usual exhaustion arguments. When f is continuous and analytic off a compact set E we can estimate its variation using less fine systems of rectangles. eXample, let n € z, Q denote the grid of closed squares of side (p + iq)2- n , p,q with vertices at the lattice points For 2-n , E; Z. Set III ,-. sup \ Vq(f) L j~l I~ JrdS . 1 the case ~: be continuous on the Riemann sphere and analytic f E.