Complete Measure - Motivation

Motivation

The need to consider questions of completeness can be illustrated by considering the problem of product spaces.

Suppose that we have already constructed Lebesgue measure on the real line: denote this measure space by (R, B, λ). We now wish to construct some two-dimensional Lebesgue measure λ2 on the plane R2 as a product measure. Naïvely, we would take the σ-algebra on R2 to be B ⊗ B, the smallest σ-algebra containing all measurable "rectangles" A₁ × A₂ for A_i ∈ B.

While this approach does define a measure space, it has a flaw. Since every singleton set has one-dimensional Lebesgue measure zero,

for "any" subset A of R. However, suppose that A is a non-measurable subset of the real line, such as the Vitali set. Then the λ2-measure of {0} × A is not defined, but

and this larger set does have λ2measure zero. So, this "two-dimensional Lebesgue measure" as just defined is not complete, and some kind of completion procedure is required.