What fails in a “lower-sum-only” extension of Riemann integration? – math.stackexchange.com

In Riemann integration, one defines both lower and upper sums $L(f,P), U(f,P),$ and declares a bounded function $f:[a,b]\to\mathbb{R}$ to be integrable if $\sup_P L(f,P)=\inf_P U(f,P).$ On the ...