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, a measure space ( X , Σ, μ ) is complete if and only if
12-517: [REDACTED] Look up completion in Wiktionary, the free dictionary. Completion may refer to: Completion (American football) Completion (oil and gas wells) Completion , a 2004 studio album by Bodychoke One of the landmarks in conveyancing , transfer of the title of property from one person to another Mathematics [ edit ] Completion (metric space) , constructing
24-517: A measure space , it has a flaw. Since every singleton set has one-dimensional Lebesgue measure zero, λ 2 ( { 0 } × A ) ≤ λ ( { 0 } ) = 0 {\displaystyle \lambda ^{2}(\{0\}\times A)\leq \lambda (\{0\})=0} for any subset A {\displaystyle A} of R . {\displaystyle \mathbb {R} .} However, suppose that A {\displaystyle A}
36-543: A confluent term rewriting system See also [ edit ] Completeness (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Completion . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Completion&oldid=1235409970 " Category : Disambiguation pages Hidden categories: Short description
48-465: A phrase the user is about to type in Knuth–Bendix completion algorithm , transforming an equation set into a confluent term rewriting system See also [ edit ] Completeness (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Completion . If an internal link led you here, you may wish to change
60-683: Is a non-measurable subset of the real line, such as the Vitali set . Then the λ 2 {\displaystyle \lambda ^{2}} -measure of { 0 } × A {\displaystyle \{0\}\times A} is not defined but { 0 } × A ⊆ { 0 } × R , {\displaystyle \{0\}\times A\subseteq \{0\}\times \mathbb {R} ,} and this larger set does have λ 2 {\displaystyle \lambda ^{2}} -measure zero. So this "two-dimensional Lebesgue measure" as just defined
72-404: Is different from Wikidata All article disambiguation pages All disambiguation pages completion [REDACTED] Look up completion in Wiktionary, the free dictionary. Completion may refer to: Completion (American football) Completion (oil and gas wells) Completion , a 2004 studio album by Bodychoke One of
84-440: 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 , Σ 0 , μ 0 ) of this measure space that is complete. The smallest such extension (i.e. the smallest σ -algebra Σ 0 ) is called the completion of the measure space. The completion can be constructed as follows: Then ( X , Σ 0 , μ 0 )
96-525: The landmarks in conveyancing , transfer of the title of property from one person to another Mathematics [ edit ] Completion (metric space) , constructing the smallest complete metric space containing a given space Construction of a complete measure space Dedekind–MacNeille completion , constructing the smallest complete lattice containing a given partial order Completion (algebra) completions in category theory Computer science [ edit ] Autocomplete , predicting
108-447: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Completion&oldid=1235409970 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Complete measure The need to consider questions of completeness can be illustrated by considering
120-645: The plane R 2 {\displaystyle \mathbb {R} ^{2}} as a product measure . Naively, we would take the 𝜎-algebra on R 2 {\displaystyle \mathbb {R} ^{2}} to be B ⊗ B , {\displaystyle B\otimes B,} the smallest 𝜎-algebra containing all measurable "rectangles" A 1 × A 2 {\displaystyle A_{1}\times A_{2}} for A 1 , A 2 ∈ B . {\displaystyle A_{1},A_{2}\in B.} While this approach does define
132-403: 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 ) . {\displaystyle (\mathbb {R} ,B,\lambda ).} We now wish to construct some two-dimensional Lebesgue measure λ 2 {\displaystyle \lambda ^{2}} on
SECTION 10
#1732765566431144-462: The smallest complete metric space containing a given space Construction of a complete measure space Dedekind–MacNeille completion , constructing the smallest complete lattice containing a given partial order Completion (algebra) completions in category theory Computer science [ edit ] Autocomplete , predicting a phrase the user is about to type in Knuth–Bendix completion algorithm , transforming an equation set into
#430569