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I am reading Elements of Integration by Bartle and I came across this:

If f is a bounded function defined on an interval $[a,b]$ and if $f$ is not too discontinuous […] In particular the lower Riemann integral of $f$ may be defined to be the supremum of the integrals of all step functions $\gamma$ such that $\gamma (x) \leq f(x)$ for all $x \in [a,b]$ and $\gamma (x)=0$ for $x \notin [a,b]$.

If it is the lower sum, should it not be the infimum?

José Carlos Santos
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Al jabra
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1 Answers1

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If $\gamma \le f$, then the integral of $\gamma$ (called a lower sum) is a lower bound for any reasonable definition of the integral of $f$. The supremum of all lower sums is in a sense the best lower bound. This is by definition the lower Riemann integral.

Umberto P.
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  • In other words, each lower sum is obtained through an infimum operation; we then take the supremum of the infima. This "maximin" or "minimax" procedure is a common situation in analysis. – Ian Dec 15 '15 at 17:33