📂Functions

Characteristic Function, Indicator Function

Definition

For $A \subset X$, the function defined as $\chi_{A} : X \to \mathbb{R}$ is referred to as the characteristic function or the indicator function.

$$\chi _{A}(x) := \begin{cases} 1, & x\in A \\ 0 ,& x \notin A \end{cases}$$

Explanation

$\chi$ is the Greek letter chi. The reason our math teacher used to say you should not write the letter x as $\chi$ but should instead use $x$ is precisely because $\chi$ is not x. Especially since it has such a strong meaning, it should not be used carelessly.

In the mathematics department, it is almost never called a characteristic function in practice, but rather read directly as [characteristic function]. It is frequently used for changing the integration range in equations involving definite integrals, for example, as follows. $$\int _{a} ^{b}f(x)g(x) dx=\int _{-\infty}^{\infty}\chi_{[a,b]}f(x)g(x)dx$$

Depending on the subject, it is also sometimes represented by the bold 1. There isn’t really a consensus on which is used more, but $\chi$ tends to have its own meaning in various fields while $\mathbf{1}$ is generally used specifically for indicator functions. Therefore, in papers rather than books, $\mathbf{1}$ seems to be used more often than $\chi$. $$\mathbf{1}_{A} = \chi _{A}(x)$$