Groups of measure-preserving homeomorphisms of noncompact 2-manifolds

Suppose M is a noncompact connected 2-manifold and μ is a good Radon measure of M with μ ( ∂ M ) = 0 . Let H ( M ) denote the group of homeomorphisms of M equipped with the compact-open topology and H ( M ) 0 denote the identity component of H ( M ) . Let H ( M ; μ ) denote the subgroup of H ( M ) consisting of μ-preserving homeomorphisms of M and H ( M ; μ ) 0 denote the identity component of H ( M ; μ ) . We use results of A. Fathi and R. Berlanga to show that H ( M ; μ ) 0 is a strong deformation retract of H ( M ) 0 and classify the topological type of H ( M ; μ ) 0 .