Set function

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In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line which consists of the real numbers and

A set function generally aims to measure subsets in some way. Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning.

Definitions[edit]

If is a family of sets over then a set function on is a function with domain and codomain or, sometimes, the codomain is instead some vector space, as with vector measures, complex measures, and projection-valued measures. The total variation of a set is

where denotes the absolute value (or more generally, it denotes the norm or seminorm if is vector-valued in a (semi)normed space). Assuming that then is called the total variation of and is called the mass of A set function is called finite if for every is finite (that is, not equal to ); every finite set function must have a finite mass.

In general, it is typically assumed that is always well-defined for all or equivalently, that does not take on both and as values. This article will henceforth assume this; although alternatively, all definitions below could instead be qualified by statements such as "whenever the sum/series is defined". This is sometimes done with subtraction, such as with the following result, which holds whenever is finitely additive:

Set difference formula: is defined with

A set is called a null set (with respect to ) or simply null if Whenever is not identically equal to either or then it is typically also assumed that:

  • null empty set: if

Common properties of set functions[edit]

A set function on is said to be[1]

  • non-negative if it is valued in
  • finitely additive if for all pairwise disjoint finite sequences such that
    • If is closed under binary unions then is finitely additive if and only if for all disjoint pairs
    • If is finitely additive and if then taking shows that which is only possible if or where in the latter case, for every (so only the case is useful).
  • countably additive or σ-additive[2] if it is finitely additive and also for all pairwise disjoint sequences in such that and where it is also required that:
    1. if is not infinite then this series must also converge absolutely, which by definition means that must be finite. This is automatically true if is non-negative.
      • As with any convergent series of real numbers, by the Riemann series theorem, the series converges absolutely if and only if its sum does not depend on the order of its terms (a property known as unconditional convergence).
    2. if is infinite then it is also required that the value of at least one of the series be finite (so that the sum of their values is well-defined). This is automatically true if is non-negative.
    • Assuming that 𝜎-additivity holds, then the series must be unconditionally convergent, which means that if is any permutation/bijection then (this is true because and also where ). This guarantees that relabeling/rearranging the sets to the new order does not affect the sum of their measures. This is a reasonable since just as the union of does not depend on the order of these sets, the same should be true of the sums and
  • a measure if is non-negative, countably additive, and has a null empty set.
  • a signed measure if is countably additive, has a null empty set, and does not take on both and as values.
  • a probability measure if it is a measure that has a mass of
  • complete if whenever then
    • Unlike most other properties, this property depends on both and 's values.
  • 𝜎-finite if there exists a sequence in such that is finite for every index and also

Arbitrary sums

As described in this article, for any family of real numbers indexed by an arbitrary indexing set it is possible to define the sum of the generalized series which as expected is denoted by (if this limit exists or diverges to ). For example, if for every then while if then converges in if and only if converges unconditionally (or equivalently, converges absolutely). It is known that if a generalized series converges in with its usual Euclidean topology (which implies that both and also converges to elements of ) then the set is necessarily countable (that is, either finite or countably infinite); this remains true if is replaced with any normed space. From it follows that where the series on the right hand side is the sum of a countable set of real numbers. Thus due to the nature of the real numbers and its topology, the definition of "countably additive" is rarely extended from countably many sets in (and the usual countable series ) to arbitrarily many sets (and the generalized series ).

Inner measures, outer measures, and other properties[edit]

A set function is said to be/satisfies[1]

  • monotone if whenever satisfy
  • modular if for all such that
    • Every finitely additive function on a field of sets is modular.
    • In geometry, real-valued valuations are modular functions.
  • submodular if for all such that
  • finitely subadditive if for all finite sequences that satisfy
  • countably subadditive if for all sequences in that satisfy
  • superadditive if whenever are disjoint with
  • continuous from above if for all non-increasing sequences of sets in such that and is finite.
  • continuous from below if for all non-decreasing sequences of sets in such that
  • infinity is approached from below if whenever satisfies then for every real there exists some such that and
  • an outer measure if is non-negative, countably subadditive, and has a null empty set.
  • an inner measure if is non-negative, superadditive, continuous from above, has a null empty set, and is approached from below.

If a binary operation is defined, then a set function is said to be

  • translation invariant if for all and such that

Topology related definitions[edit]

If is a topology on then a set function is said to be:

  • -additive if whenever is directed with respect to and satisfies
    • is directed with respect to if and only if it is not empty and for all there exists some such that and
  • inner regular or tight if for every
  • outer regular if for every
  • regular if it is both inner regular and outer regular.
  • locally finite if for every point there exists some neighborhood of this point such that is finite.
  • Radon measure if it is a regular and locally finite measure.

Relationships between set functions[edit]

If are two set functions over then:

  • is said to be absolutely continuous with respect to or dominated by , written if for every set that belongs to the domain of both if then
    • If and are -finite measures on the same measurable space and if then the Radon–Nikodym derivative exists and for every measurable
  • are singular, written if there exist disjoint sets and in the domains of such that for all in the domain of and for all in the domain of

Examples[edit]

Examples of set functions include:

  • The function
    assigning densities to sufficiently well-behaved subsets is a set function.
  • The Lebesgue measure is a set function that assigns a non-negative real number to any set of real numbers, that is in Lebesgue -algebra.[3]
  • A probability measure assigns a probability to each set in a σ-algebra. Specifically, the probability of the empty set is zero and the probability of the sample space is with other sets given probabilities between and
  • A possibility measure assigns a number between zero and one to each set in the powerset of some given set. See possibility theory.
  • A random set is a set-valued random variable. See the article random compact set.

Properties[edit]

Extending set functions from a semialgebra[edit]

Suppose that is a set function on a semialgebra over and let

which is the algebra on generated by The archetypal example of a semialgebra that is not also an algebra is the family
on where for all [4] Importantly, the two non-strict inequalities in cannot be replaced with strict inequalities since semialgebras must contain the whole underlying set that is, is a requirement of semialgebras (as is ).

If is finitely additive then it has a unique extension to a set function on defined by sending (so these are pairwise disjoint) to:[4]

This extension will also be finitely additive: for any pairwise disjoint [4]

If in addition is extended real-valued and monotone (which, in particular, will be the case if is non-negative) then will be monotone and finitely subadditive: for any such that [4]

See also[edit]

Notes[edit]

  1. ^ a b Durrett 2019, pp. 1–37, 455–470.
  2. ^ Durrett 2019, pp. 466–470.
  3. ^ Kolmogorov and Fomin 1975
  4. ^ a b c d Durrett 2019, pp. 1–9.

References[edit]

  • Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020.
  • Kolmogorov, Andrey; Fomin, Sergei V. (1957). Elements of the Theory of Functions and Functional Analysis. Dover Books on Mathematics. New York: Dover Books. ISBN 978-1-61427-304-2. OCLC 912495626.
  • A. N. Kolmogorov and S. V. Fomin (1975), Introductory Real Analysis, Dover. ISBN 0-486-61226-0
  • Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.

Further reading[edit]