The upper sums are bounded below by every lower sum, so they have an infimum; the lower sums are bounded above, so they have a supremum. Those two numbers are the upper and lower integrals, and they always exist for a bounded function. The function is called integrable when they are equal, and the common value is the integral. Nothing here assumes they are equal, and for some functions they are not.

Two auctions for the same field, one of buyers bidding up and one of sellers coming down. The highest bid and the lowest asking price always exist, and they always satisfy the obvious inequality. A sale happens exactly when they meet.

On each piece of the partition one bar is as tall as the function ever gets and the other as short. The upper bars overstate the area and the lower bars understate it, and the whole chapter is about whether the two totals can be brought together.

For bounded $f$ on $[a,b]$ define the upper integral $\overline{\int_a^b}f=\inf_P U(f,P)$ and the lower integral $\underline{\int_a^b}f=\sup_P L(f,P)$. Both exist by completeness, and $\underline{\int}f\le\overline{\int}f$ always. $f$ is Riemann integrable when the two are equal, and the common value is $\int_a^b f$.

• Both integrals exist for every bounded function, by completeness applied to the two separated families of sums.
• Integrability is the extra condition that they agree. It is not automatic.
• The function that is $1$ at every rational and $0$ at every irrational has upper integral $b-a$ and lower integral $0$ on $[a,b]$, so it is not integrable — every piece contains both kinds of number.

Fix one partition on the upper side. Every lower sum is below the upper sum of that partition, by the common refinement argument of the last lesson, so that upper sum is an upper bound for the whole family of lower sums — and the lower integral, being the least such bound, is at or below it. That holds for every partition on the upper side, so the lower integral is a lower bound for the whole family of upper sums, and the upper integral, being the greatest such bound, is at or above it.

Fix one partition and consider its upper sum.

Every lower sum is below it, by the common refinement.

So it bounds the lower sums above, and the supremum is the least such bound.

Nothing about Q was used, so the lower integral bounds every upper sum below.

And the upper integral is the greatest lower bound of the upper sums.

The two integrals always exist. Integrability is whether they agree.

• The upper and lower integrals always exist for a bounded function.
• Integrability is the two being equal, and it is not automatic.
• Every upper sum bounds the integral above and every lower sum below.
• The function that is one on the rationals is bounded and not integrable.

Great work! You now know how to:

• Define the upper and lower integrals and say why both exist
• State what integrability is and why it is an extra condition
• Give a bounded function that is not integrable
• Use any partition to bound an integral