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Suppose M is a noncompact connected n-manifold and ω is a good Radon measure of M with ω(∂M) = 0. Let ℋ(M,ω) denote the group of ω-preserving homeomorphisms of M equipped with the compact-open topology, and ${\mathscr{H}}_E(M,\omega )$ the subgroup consisting of all h ∈ ℋ(M,ω) which fix the ends of M. S. R. Alpern and V. S. Prasad introduced the topological vector space (M,ω) of end charges of M and the end charge homomorphism $c^{\omega }:{\mathscr{H}}_E(M,\omega )\to (M,\omega )$, which measures for each $h\in {\mathscr{H}}_E(M,\omega )$ the mass flow toward ends induced by h. We show that the map $c^{\omega }$ has...