On closability of classical Dirichlet forms

We construct a measure μ on R2 such that the classical Dirichlet form E(f,g)=∫(∇f,∇g) dμ is closable, but the partial Dirichlet form Ex(f,g)=∫∂xf∂xg dμ is not. This proves the well-known conjecture of M. Röckner.