Tagged partition

A tagged partition of $[a,b]$ is a pair $(P,T)$ where:

• $P:a=x_0<\cdots<x_n=b$ is a partition, and
• $T=(t_1,\dots,t_n)$ is a choice of tags with $t_i\in[x_{i-1},x_i]\quad\text{for each }i=1,\dots,n.$

Tagged partitions specify where a function is sampled to form Riemann sums. Different tagging conventions (e.g., left endpoints, right endpoints, midpoints) yield different approximations for finite partitions.

Examples:

• Left-endpoint tags: $t_i=x_{i-1}$ for all $i$.
• Right-endpoint tags: $t_i=x_i$ for all $i$.
• Midpoint tags: $t_i=\frac{x_{i-1}+x_i}{2}$.