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Although $K$ is a compact Hausdorff topological group in its $\\tau$-topology, the inclusion of $K$ into $E(X,G)$ need not be Borel. We show that normalized Haar measure on $K$ nevertheless determines, via the Riesz--Markov theorem, a canonical regular Borel probability measure $\\mu_K$ on $E(X,G)$, called its Haar decompression.\n  Our principal structural result states that, for every tame flow, $(E(X,G),G)$ is amenable if and only if $(X,G)$ is hereditarily amenable. 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