Complete measure

In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, (X, Σ, μ) is complete if and only if

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 λ²-measure zero. So, this "two-dimensional Lebesgue measure" as just defined is not complete, and some kind of completion procedure is required.

Given a (possibly incomplete) measure space (X, Σ, μ), there is an extension (X, Σ₀, μ₀) of this measure space that is complete. The smallest such extension (i.e. the smallest σ-algebra Σ₀) is called the completion of the measure space.

The completion can be constructed as follows:

Then (X, Σ₀, μ₀) is a complete measure space, and is the completion of (X, Σ, μ).

...
Wikipedia