Editing Complete measure (section)

==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 <math>(\R, B, \lambda).</math> We now wish to construct some two-dimensional Lebesgue measure <math>\lambda^2</math> on the plane <math>\R^2</math> as a [[product measure]]. Naively, we would take the [[Sigma algebra|{{sigma}}-algebra]] on <math>\R^2</math> to be <math>B \otimes B,</math> the smallest {{sigma}}-algebra containing all measurable "rectangles" <math>A_1 \times A_2</math> for <math>A_1, A_2 \in B.</math>

While this approach does define a [[measure space]], it has a flaw. Since every [[Singleton (mathematics)|singleton]] set has one-dimensional Lebesgue measure zero,
<math display=block>\lambda^2(\{0\} \times A) \leq \lambda(\{0\}) = 0</math>
for {{em|any}} subset <math>A</math> of <math>\R.</math> However, suppose that <math>A</math> is a [[Non-measurable set|non-measurable subset]] of the real line, such as the [[Vitali set]]. Then the <math>\lambda^2</math>-measure of <math>\{0\} \times A</math> is not defined but
<math display=block>\{0\} \times A \subseteq \{0\} \times \R,</math>
and this larger set does have <math>\lambda^2</math>-measure zero. So this "two-dimensional Lebesgue measure" as just defined is not complete, and some kind of completion procedure is required.