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A. Baernstein II ( Comparison of      p p\mspace {1mu}      -harmonic measures of subsets of the unit circle , St. Petersburg Math. J.  9  (1998), 543–551, p. 548), posed the following question: If      G G       is a union of      m m       open arcs on the boundary of the unit disc      D \mathbf {D}      , then is      ω  a , p ( G ) = ω  a , p ( G ¯  ) \omega _{a,p}(G)=\omega _{a,p}(\overline {G})      , where      ω  a , p \omega _{a,p}       denotes the      p p\mspace {1mu}      -harmonic measure? (Strictly speaking he stated this question for the case      m = 2 m=2      .) For      p = 2 p=2       the positive answer to this question is well known. Recall that for      p ≠  2 p \ne 2       the      p p\mspace {1mu}      -harmonic measure, being a nonlinear analogue of the harmonic measure, is not a measure in the usual sense. The purpose of this note is to answer a more general version of Baernstein’s question in the affirmative when      1 &gt; p &gt; 2 1&gt;p&gt;2      . In the proof, using a deep trace result of Jonsson and Wallin, it is first shown that the characteristic function      χ  G \chi _G       is the restriction to      ∂  D \partial \mathbf {D}       of a Sobolev function from      W 1 , p ( C ) W^{1,p}(\mathbf {C})      . For      p ≥  2 p \ge 2       it is no longer true that      χ  G \chi _G       belongs to the trace class. Nevertheless, we are able to show equality for the case      m = 1 m=1       of one arc for all      1 &gt; p &gt; ∞  1&gt;p&gt;\infty      , using a very elementary argument. A similar argument is used to obtain a result for starshaped domains. Finally we show that in a certain sense the equality holds for almost all relatively open sets.

A problem of Baernstein on the equality of the 𝑝\mspace{1𝑚𝑢}-harmonic measure of a set and its closure