Supremum/infimum definition

$$\underline{I_{a}^b}(f) = \sup \{ L_{P}(f) |P \text{ is a partition } of [a,b] \}$$

$$\overline{I_{a}^b}(f) = \inf \{ U_{P}(f) |P \text{ is a partition } of [a,b] \}$$

Limit definition

Upper Integral

The Upper Integral $\overline{I_{a}^b}(f)$ is the number $I$ that the upper sums converge to as the norm of the partition approaches zero. Formally: $$\lim_{ ||P|| \to 0 } U_{P}(f) = I$$ or by $\epsilon-\delta$ definition:

$$\forall\epsilon > 0,\exists\delta > 0 \text{ s.t. } \forall P \text{ with } ||P|| < \delta \implies |U_{P}(f) - I| < \epsilon$$

Lower Integral

The Lower Integral #TODO

Order of Equality

For any single partition $P$:

$$ L_{p}(f) \le \underline{I_{a}^b} \le \overline{I_{a}^b} \le U_{p}(f) $$

The actual area (if it exists) is always sandwiched between the Lower and Upper integrals, which themselves are sandwiched between any specific Lower or Upper sum.