The definition of integral defines lower integral of $f$ on $[a,b]$, $\underline{I}_a^b (f)$, as the supremum of the lower sums of all partitions. But isn't the lower value of $f(x)$ infimum? how does this definition work?
2026-04-04 02:32:08.1775269928
How does the superimum of lower sums of partitions work?
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This is an example of "minimax" or "maximin": you have a function of two variables*. Taking the min (resp. max) in one variable gives a function of one variable. You then take the max (resp. min) in that variable and you wind up with a number. This is a fairly common notion.
It's the right thing to do in integration because you can imagine drawing the step function which corresponds to the lower sum of your function with a particular partition. When you refine the partition, part of the step function increases, getting closer to the actual graph in the process. The supremum operation in your definition is responsible for this "increasing" effect.
* It's not exactly a function of two variables in integration: the minimum is taken over evaluation points, while the maximum is taken over partitions. But the evaluation points are not independent of the partition. Still, this is the idea.