Neha Patil (Editor)

Characteristic function (convex analysis)

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In the field of mathematics known as convex analysis, the characteristic function of a set is a convex function that indicates the membership (or non-membership) of a given element in that set. It is similar to the usual indicator function, and one can freely convert between the two, but the characteristic function as defined below is better-suited to the methods of convex analysis.

Contents

Definition

Let X be a set, and let A be a subset of X . The characteristic function of A is the function

χ A : X → R ∪ { + ∞ }

taking values in the extended real number line defined by

χ A ( x ) := { 0 , x ∈ A ; + ∞ , x ∉ A .

Relationship with the indicator function

Let 1 A : X → R denote the usual indicator function:

1 A ( x ) := { 1 , x ∈ A ; 0 , x ∉ A .

If one adopts the conventions that

  • for any a ∈ R ∪ { + ∞ } , a + ( + ∞ ) = + ∞ and a ( + ∞ ) = + ∞ ;
  • 1 0 = + ∞ ; and
  • 1 + ∞ = 0 ;

    • then the indicator and characteristic functions are related by the equations

    1 A ( x ) = 1 1 + χ A ( x )

    and

    χ A ( x ) = ( + ∞ ) ( 1 − 1 A ( x ) ) .

    References

    Characteristic function (convex analysis) Wikipedia
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