Lower sum

A lower sum of a bounded function $f:[a,b]\to\mathbb R$ with respect to a partition $P=\{x_0,\dots,x_n\}$ is the number

where $m_i=\inf\{f(x):x\in[x_{i-1},x_i]\}$ is the infimum of $f$ on the $i$th subinterval.

Lower sums are paired with upper sums to define Riemann integrability via the gap $U(f,P)-L(f,P)$.

Examples:

• For $f(x)=x$ on $[0,1]$ and $P=\{0,\tfrac12,1\}$, one gets $L(f,P)=0\cdot\tfrac12+\tfrac12\cdot\tfrac12=\tfrac14$.
• If $f(x)=c$ is constant on $[a,b]$, then $L(f,P)=c(b-a)$ for every partition $P$.