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

Step integral from above

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

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

denote a function. For a step function from above

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

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

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

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