Non-Archimedean (uniformly) continuous measures on homogeneous spaces. (English) Zbl 0541.46062

Let G be a locally compact zerodimensional group with a closed subgroup H. Let K be a nonarchimedean valued complete field. If is assumed that G has a K-valued Haar measure. In a standard way one defines a topology, a uniformity and quasi-invariant measures on G/H. Let \(BC(G/H)\quad resp.\quad BU\subset(G/H),\quad C_{\infty}(G/H))\) be the space of all continuous functions G/\(H\to K\) that are bounded (resp. bounded and uniformly continuous, zero at infinity), with the uniform norm. Let \(M^{\infty}(G/H)\) be the dual space of \(C_{\infty}(G/H)\). \(M^{\infty}(G/H)\) is isomorphic to the space of all measures on G/H. The weak topology on \(M^{\infty}(G/H)\) (a slightly confusing term in this context) is the topology induced by the seminorms \(\mu \mapsto N_{\mu}(x)\), where \(N_{\mu}(x)=\inf_{A\ni x}\sup_{B\subset A}| \mu(B)|\) (\(x\in G/H).\)
The following is proved. Let \(\mu\) be a quasi-invariant measure on G/H. The map \(f\to f\mu\) from \(B\cup C(G/H)\) (resp. \(B\subset(G/H)\) with pointwise topology) to \(M^{\infty}(G/H)\) (resp. \(M^{\infty}(G/H)\) with the weak topology) is a linear homeomorphism onto a closed subspace. The latter consists precisely of those measures that translate continuously (continuously for the weak topology).
Remark. The case \(H=\{e\}\) has been treated by the reviewer [Indag. Math. 33, 78-85 (1971; Zbl 0216.412), Theorem 4.1].
Reviewer: W.Schikhof


46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46E27 Spaces of measures
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
43A05 Measures on groups and semigroups, etc.


Zbl 0216.412
Full Text: Numdam EuDML


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