Local Field - Induced Absolute Value

Induced Absolute Value

Given a locally compact topological field K, an absolute value can be defined as follows. First, consider the additive group of the field. As a locally compact topological group, it has a unique (up to positive scalar multiple) Haar measure μ. The absolute value is defined so as to measure the change in size of a set after multiplying it by an element of K. Specifically, define |·| : K → R by

for any measurable subset X of K (with 0 < μ(X) < ∞). This absolute value does not depend on X nor on the choice of Haar measure (since the same scalar multiple ambiguity will occur in both the numerator and the denominator).

Given such an absolute value on K, a new induced topology can be defined on K. This topology is the same as the original topology. Explicitly, for a positive real number m, define the subset B_m of K by

Then, the B_m make up a neighbourhood basis of 0 in K.