Vitushkin’s Conjecture for Removable Sets by James J. Dudziak

Author:James J. Dudziak
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

From we see that

Thus we may apply [RUD, 11.29] and [RUD, 6.19] as in the proof of Frostman’s Lemma (2.9) to conclude that there exists a sequence decreasing to 0 such that the sequence converges weakly to a continuous linear functional Λ on represented by a regular complex Borel measure μ on B. This means that for any ,

Since each Λ ε annihilates those continuous functions whose support is a distance more that ε from K, it follows from that Λ annihilates those continuous functions compactly supported in . Thus by [RUD, 6.19 (2) and 3.17], , i.e., μ is supported on K.

Given , use Tietze’s Extension Theorem [RUD, 20.4] to get a continuous function ϕ such that for all ζ whose distance to K is at most half the distance of z to K. By Lemma 2.13, for every which is less than half the distance of z to K, we then have

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