Bounded interval/Real function/Lower integral as supremum of step integrals from below/Definition

Lower integral

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

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

denote a function, which is bounded from below. Then the supremum of all step integrals of step functions from below of ${}f$ is called the lower integral of ${}f$.