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

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