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
>
p
>
2
1>p>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
>
p
>
∞
1>p>\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.