Complete Measure - Construction of A Complete Measure

Construction of A Complete Measure

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:

  • let Z be the set of all subsets of μ-measure zero subsets of X (intuitively, those elements of Z that are not already in Σ are the ones preventing completeness from holding true);
  • let Σ0 be the σ-algebra generated by Σ and Z (i.e. the smallest σ-algebra that contains every element of Σ and of Z);
  • there is a unique extension μ0 of μ to Σ0 given by the infimum

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

In the above construction it can be shown that every member of Σ0 is of the form A ∪ B for some A ∈ Σ and some B ∈ Z, and

Read more about this topic:  Complete Measure

Famous quotes containing the words construction of, construction, complete and/or measure:

    “There is, I think, no point in the philosophy of progressive education which is sounder than its emphasis upon the importance of the participation of the learner in the formation of the purposes which direct his activities in the learning process, just as there is no defect in traditional education greater than its failure to secure the active cooperation of the pupil in construction of the purposes involved in his studying.”
    —John Dewey (1859–1952)

    “There’s no art
    To find the mind’s construction in the face.”
    —William Shakespeare (1564–1616)

    “‘Tis chastity, my brother, chastity.
    She that has that is clad in complete steel,
    And like a quivered nymph with arrows keen
    May trace huge forests and unharbored heaths,
    Infamous hills and sandy perilous wilds,
    Where, through the sacred rays of chastity,
    No savage fierce, bandit, or mountaineer
    Will dare to soil her virgin purity.”
    —John Milton (1608–1674)

    “One might imagine that a movement which is so preoccupied with the fulfillment of human potential would have a measure of respect for those who nourish its source. But politics make strange bedfellows, and liberated women have elected to become part of a long tradition of hostility to mothers.”
    —Elaine Heffner (20th century)