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