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.

Definition[edit]

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

\chi_{A} : X \to \mathbb{R} \cup \{ + \infty \}

taking values in the extended real number line defined by

\chi_{A} (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases}

Relationship with the indicator function[edit]

Let \mathbf{1}_{A} : X \to \mathbb{R} denote the usual indicator function:

\mathbf{1}_{A} (x) := \begin{cases} 1, & x \in A; \\ 0, & x \not \in A. \end{cases}

If one adopts the conventions that

  • for any a \in \mathbb{R} \cup \{ + \infty \}, a + (+ \infty) = + \infty and a (+\infty) = + \infty;
  • \frac{1}{0} = + \infty; and
  • \frac{1}{+ \infty} = 0;

then the indicator and characteristic functions are related by the equations

\mathbf{1}_{A} (x) = \frac{1}{1 + \chi_{A} (x)}

and

\chi_{A} (x) = (+ \infty) \left( 1 - \mathbf{1}_{A} (x) \right).

Bibliography[edit]