The author exhibits a measure
on the plane such that the Dirichlet form
is closable, whereas the form
is not. This gives a positive answer to a question of S. Albeverio
and M. Röckner
[J. Funct. Anal. 88, No. 2, 395–436 (1990; Zbl 0737.46036
)]. The measure
is restriction of Lebesgue measure to an open subset of the unit square and the construction is based on a Cantor set of positive measure.