Characteristic function (convex analysis)

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

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

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

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