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**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].

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

### MSC:

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. |

### Keywords:

homogeneous space; locally compact zerodimensional group; uniformly continuous measure; continuous measure; nonarchimedean valued complete field; quasi-invariant measures; weak topology; measures that translate continuously### Citations:

Zbl 0216.412### References:

[1] | A.C.M. Van Rooij : Non-archimedean Functional Analysis , Marcel Dekker, Inc., New York and Basel (1978). · Zbl 0396.46061 |

[2] | N. Bourbaki : XXIX, Elem. de Math., Livre IV , IntĂ©gration chap. 7 et 8, Hermann, Paris (1954). |

[3] | L. Duponcheel : Non-archimedean Induced Representations and Related Topics , Thesis (1979). |

[4] | L. Duponcheel : Non-archimedean quasi-invariant measures on homogeneous spaces . Indag. Math., Volumen 45, Fasciculus 1 (1983). · Zbl 0513.43002 |

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