Ultrafilter - Formal Definition

Formal Definition

Given a set X, an ultrafilter on X is a set U consisting of subsets of X such that

  1. The empty set is not an element of U
  2. If A and B are subsets of X, A is a subset of B, and A is an element of U, then B is also an element of U.
  3. If A and B are elements of U, then so is the intersection of A and B.
  4. If A is a subset of X, then either A or X \ A is an element of U. (Note: axioms 1 and 3 imply that A and X \ A cannot both be elements of U.)

A characterization is given by the following theorem. A filter U on a set X is an ultrafilter if any of the following conditions are true:

  1. There is no filter F finer than U, i.e., implies U = F.
  2. implies or .
  3. or .

Another way of looking at ultrafilters on a set X is to define a function m on the power set of X by setting m(A) = 1 if A is an element of U and m(A) = 0 otherwise. Then m is a finitely additive measure on X, and every property of elements of X is either true almost everywhere or false almost everywhere. Note that this does not define a measure in the usual sense, which is required to be countably additive.

For a filter F which is not an ultrafilter, one would say m(A) = 1 if A ∈ F and m(A) = 0 if X \ A ∈ F, leaving m undefined elsewhere.

Read more about this topic:  Ultrafilter

Famous quotes containing the words formal and/or definition:

    “It is in the nature of allegory, as opposed to symbolism, to beg the question of absolute reality. The allegorist avails himself of a formal correspondence between “ideas” and “things,” both of which he assumes as given; he need not inquire whether either sphere is “real” or whether, in the final analysis, reality consists in their interaction.”
    —Charles, Jr. Feidelson, U.S. educator, critic. Symbolism and American Literature, ch. 1, University of Chicago Press (1953)

    “Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
    —Elaine Heffner (20th century)