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We answer the question for the specific situation of non-Archimedean Levi-Civita vector spaces and show that they behave in the same manner as in the real case. To this end, we develop a Lebesgue measure in these spaces that is invariant under affine transformations and satisfies commonly expected properties of Lebesgue measures, and in particular a substitution rule based on Jacobians of transformations. Using the tools from this measure theory, we will show that the Obtuse Angle Theorem also holds on the non-Archimedean Levi-Civita vector spaces.
M. Berz, S. Troncoso, AMS Contemporary Mathematics 596 (2013) 1-21
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