Let \([a,b]\) be a closed real interval and let \(f:[a,b]\mapsto\mathbb R\) be an arbitrary bounded function. Then the following integrals exist and are well-defined:
The Riemann upper integral of \(f\) over the interval \([a,b]\) is the infimum of all possible Riemann Integrals for step functions \(\phi\) such that their values are bounded below by the respective values of \(f\), formally:
\[\int_{a}^{b~*}f(x)dx:=\inf\left\{\int_{a}^b\phi(x)dx:~~\phi(x)\ge f(x)\text{ for all }x\in[a,b]\text{ and }\phi\text{ is a step function over }[a,b]\right\}.\]
The Riemann lower integral of \(f\) over the interval \([a,b]\) is the supremum of all possible Riemann Integrals for step functions \(\phi\) such that their values are bounded above by the respective values of \(f\), formally:
\[\int_{a~*}^bf(x)dx:=\sup\left\{\int_{a}^b\phi(x)dx:~~\phi(x)\le f(x)\text{ for all }x\in[a,b]\text{ and }\phi\text{ is a step function over }[a,b]\right\}.\]
In the following interactive figure, an example bounded function on the closed interval \([-3,3]\) is generated. You can drag the red points to change the function. The slider to the right changes the step functions of the upper (red) and lower (green) Riemann integrals of the corresponding step functions. The infimum and supremum of the upper and lower integrals of step functions are reached, if the step length goes to zero (i.e. number of steps in the closed interval goes to infinity).
Proofs: 1
Definitions: 1
Examples: 2
Lemmas: 3
Proofs: 4 5 6