Upper & Lower integral

Let $\mathbb{P}$ be the set of all possible partitions of the interval $[a, b]$.

Upper Integral

$$U(f)=\inf{\set{U(f;P);P\in\mathbb{P}}}=\overline{\int_{a}^{b}{f}}$$

Lower Integral

$$L(f)=\sup{\set{L(f;P);P\in\mathbb{P}}}=\underline{\int_{a}^{b}{f}}$$

For a bounded function $f$, always $L(f)\le U(f)$