Definition

Let $a

Note: Continuity and boundedness are sufficient conditions that guarantee integrability (this is the typical assumption in MAT137), but the definition of the integral only requires boundedness.

Let the Upper and Lower Integrals be

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

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

$$\underline{I_a^b}(f) = \overline{I_{a}^b}(f)$$

Then the definite integral is

$$\int_{a}^b f(x) \, dx = \underline{I_{a}^b}(f) = \overline{I_{a}^b}(f)$$

$\epsilon$-characterisation of Integrability

Let $a

$$\forall \epsilon > 0, \exists \text{a partition } P \text{ of } [a,b] \text{ s.t } U_{p}(f) - L_{p}(f) < \epsilon$$

This is useful (particularly for MAT237) because

• You don't have to compute the lower and upper integrals
• You only need to show that the upper and lower sums can be made arbitrarily close
• This is often easier for proving integrability