Lower sum (Riemann)

A weighted sum of infima of f over subintervals of a partition.

Lower sum (Riemann)

Let $f:[a,b]\to\mathbb{R}$ be bounded and let $P:a=x_0<\cdots<x_n=b$ be a partition. For each subinterval, define

m_i := \inf\{f(x): x\in[x_{i-1},x_i]\}.

The lower sum of $f$ with respect to $P$ is

L(f,P) := \sum_{i=1}^n m_i\, (x_i-x_{i-1}).

Lower sums approximate the integral from below. As the partition is refined, lower sums increase (or stay the same).

Examples:

If $f(x)=x$ on $[0,1]$ and $P:0<1/2<1$ , then $m_1=0$ , $m_2=1/2$ , so $L(f,P)=0\cdot\frac12+\frac12\cdot\frac12=\frac14$ .
If $f(x)=c$ is constant, then $L(f,P)=c(b-a)$ for every $P$ .
For $f=\mathbf{1}_{\mathbb{Q}}$ on $[0,1]$ , every subinterval has $m_i=0$ and $M_i=1$ , so $L(f,P)=0$ and $U(f,P)=1$ for all $P$ .