quality:correction
Definition and first consequences

A function f: R → R {\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} } is called a step function if it can be written as

where n ≥ 0 {\displaystyle n\geq 0}, α i {\displaystyle \alpha _{i}} are real numbers, A i {\displaystyle A_{i}} are intervals, and χ A {\displaystyle \chi _{A}} is the indicator function of A {\displaystyle A}:

In this definition, the intervals A i {\displaystyle A_{i}} can be assumed to have the following two properties:

Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold. For example, the step function

can be written as