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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 Reimann 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 not in [a,b]

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

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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.

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  • $\begingroup$ 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. $\endgroup$ – Ian Dec 15 '15 at 17:33

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