indicator function

let $\text{[math]}$ be a bounded subset. the indicator function $\text{[math]}$ is
$\text{[math]}$ [cite:;taken from @calc_hubbard_2015 definition 4.1.1 (indicator function)]

the term "characteristic function" has an unrelated meaning in classic probability theory. for this reason, traditional probabilists use the term indicator function for the function defined here almost exclusively, while mathematicians in other fields are more likely to use the term characteristic function to describe the function that indicates membership in a set.
https://en.wikipedia.org/wiki/Indicator_function

let $\text{[math]}$ be a set, the function $\text{[math]}$ that is defined for $\text{[math]}$ , such that $\text{[math]}$ for every $\text{[math]}$ and $\text{[math]}$ for every $\text{[math]}$ , here, $\text{[math]}$ is called the characteristic function of $\text{[math]}$ .

$\text{[math]}$

we define the indicator function on the set $\text{[math]}$ by
$\text{[math]}$ [cite:;taken from @klenke_prob_2020 remark 1.14]