Bounded interval/Real Function/Step integral from below/Definition

Step integral from below

Let ${}I$ denote a bounded interval, and let

$f\colon I\longrightarrow \mathbb {R}$

denote a function. For a step function from below

$t\colon I\longrightarrow \mathbb {R}$

of ${}f$, with respect to the partition ${}a_{i}$, ${}i=0,\ldots ,n$, and values ${}s_{i}$, ${}i=1,\ldots ,n$, the step integral

${}S:=\sum _{i=1}^{n}s_{i}{\left(a_{i}-a_{i-1}\right)}\,$

is called a step integral from below for ${}f$ on ${}I$.