P-adic integration and Hartogs-Stawski's theorem

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The purpose of this reseach is to give a correct proof of Stawski's Theorem (non-archimedean version of Hartogs' Theorem) . The first step of the resarch is to put in order Stawski's outer linear measure theory. This was done by a good advice of Alain Escassut.A correct proof of Hartogs-Stawski's theorem was given when the underling field K is a complete, but not locally compact subfield of the p-adic complex field CィイD2pィエD2. If the value group |KィイD1xィエD1| is discrete, the following revised version of Stawski's theorem holds.Theorem 1 If a function f (x) = f(xィイD21ィエD2, xィイD22ィエD2, ..., xィイD2nィエD2) is analytic for each variable on the domain|xィイD21ィエD2|≦ RィイD21ィエD2,|xィイD22ィエD2|≦ RィイD22ィエD2,..., |xィイD2nィエD2|≦ RィイD2nィエD2then the function f(x) is an analytic function in the whole variables on the domain|xィイD21ィエD2|< RィイD21ィエD2,|xィイD22ィエD2|≦ ィイD2qィエD2RィイD22ィエD2,..., |xィイD2nィエD2|≦ ィイD2qィエD2RィイD2nィエD2where q = |π|< 1 (πis a prime element of K) .By symmetry we slightly extended the domain of analyticity of the functions. We also see that the Theorem holds if the field K is a general non-archimedean field which is complete, but not locally compact.