4 MARTIN BERZ AND SEBASTIAN TRONCOSO
order topology, and on the original real points, their derivatives agree with those
of the underlying original function. The subject appeared again in the work by
Ostrowski [23], Neder [22], and later in the work of Laugwitz [15]. Two more
recent accounts of this work can be found in the book by Lightstone and Robinson
[19], which ends with the proof of Cauchy completeness, as well as in Laugwitz’
account on Levi-Civita’s work [16], which also contains a summary of properties of
Levi-Civita fields.
In [2] it is shown by explicit construction that C is algebraically closed, and that
R is real-closed, which also follows from general valuation theory, and specifically
for example the work of Rayner[26]. Compared to the general Hahn fields, the Levi-
Civita fields are characterized by well-ordered exponent sets that are particularly
”small”, and indeed minimally small to allow simultaneously algebraic closure and
the Cauchy completeness, as shown in [4].
2. Measure Theory on Levi-Civita vector spaces
Attempts to formulate meaningful measure theories on non-Archimedean fields
have to necessarily follow modified approaches rather than the common method of
Lebesgue. The total disconnectedness of these spaces under the order topology, the
lack of existence of suprema and infima of bounded sets, and the different orders
of magnitude that exist in the non-Archimedean structures, prevent the use of
concepts of measure theory for example on Banach spaces. We begin our discussion
with several general observations related to the introduction of measures on Levi-
Civita vector spaces. First we establish that the situation is indeed fundamentally
different from the real case.
Proposition 2.1. There is no non-trivial translation invariant measure on the
Levi-Civita field R or the vector spaces
Rd.
Proof. We follow an indirect argument. Suppose there is a non-trivial mea-
sure m in R. Let A R be any bounded set with non-vanishing measure. Let
b R be a bound of A, i.e. x A |x| b. Now consider the family of translates
of A as follows:
An = n ·
b
δ
+ A
where δ is an arbitrary positive infinitely small number. By translation invariance
we have m(An) = m(A), and by the boundedness of A by b, we also have Ai∩Aj =
for i = j. Now consider the set
B = [−
b
δ
, +
b
δ
].
Apparently we have An B for all n N. Because of the Archimedicity of R and
because m(B) is finite due to the measure being non-trivial, there exists k N such
that k · m(A) m(B), so we have on the one hand
m
n
i=1
Ai m(B)
but on the other hand we also have
n
i=1
Ai B
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