Extension of measures to infinite-dimensional spaces over $p$-adic field

In carrying out analysis on infinite dimensional spaces over /7-adics, it is useful to give integral representations of functions. Satoh considered a normed vector space H over a local field K with orthonormal Schauder basis ([14]). He showed that any admissible probability measure on K is extended to a measure on the completion of H with respect to a measurable norm, applying Prokhorov's measure extension theorem to the projective limit of the images of orthogonal projections on H. This can be applied to a space of polynomials with coefficients in /7-adics. On the other hand the present paper aims at extending probability measures to spaces including extension fields over /7-adics of infinite degree, in which there exist no orthonormal basis in the sense of [14], except the case of unramified extensions. The spaces to which we extend measures are completions of infinite extension fields over /7-adics with respect to specific seminorms induced by projections naturally related with traces on subextensions. We notice that our projections are not necessarily orthogonal in the sense of [14]. The subjects of our theorem include for instance the algebraic closure and the maximal unramified extension of the /7-adic field. Kochubei proved independently that Gaussian measures on a local field can be extended to completion of an infinite extension and constructed a fractional differentiation operator relative to the measure ([9]). Let p be a fixed prime integer. The /7-adic field Qp consists of formal power series

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