Step Function - Definition and First Consequences

Definition and First Consequences

A function is called a step function if it can be written as

for all real numbers

where are real numbers, are intervals, and is the indicator function of :

\chi_A(x) =
\begin{cases}
1 & \mbox{if } x \in A, \\
0 & \mbox{if } x \notin A. \\
\end{cases}

In this definition, the intervals can be assumed to have the following two properties:

  1. The intervals are disjoint, for
  2. The union of the intervals is the entire real line,

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

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