Haar decompression and amenability of Ellis flows

Let $(X,G)$ be a tame flow and let $K$ be an Ellis group of its enveloping semigroup $E(X,G)$. 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. 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. For a tame hereditarily amenable flow, every Haar decompression is $G$-invariant whenever $X$ is metrizable or $G$ is countable. For metrizable minimal tame flows admitting an invariant measure, the evaluation pushforward of every Haar decompression at every point is the unique invariant measure. Moreover, every ergodic invariant measure on a metrizable tame flow has minimal support; consequently, every ergodic invariant measure on a metrizable tame ambit is obtained by evaluating a suitable Haar decompression.