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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 : ==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 ''λ''2 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" ''A''1 × ''A''2 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 ''λ''2-measure of × ''A'' is not defined, but : and this larger set does have ''λ''2-measure zero. So, this "two-dimensional Lebesgue measure" as just defined is not complete, and some kind of completion procedure is required. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Complete measure」の詳細全文を読む スポンサード リンク
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