Bounded interval/Real function/Upper integral as infimum of step integrals from above/Definition

Upper integral

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

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

denote a function, which is bounded from above. Then the infimum of all step integrals of step functions from above of ${}f$ is called the upper integral of ${}f$.