Refinement of a partition

A refinement of a partition $P$ of $[a,b]$ is a partition $Q$ of $[a,b]$ such that every point of $P$ is also a point of $Q$. In other words, $P\subseteq Q$ in the sense of subset .

Refinements are used to compare upper sums and lower sums : making a partition finer is the basic way to force these sums closer together in Riemann integrability .

Examples:

• If $P=\{0,1\}$ and $Q=\{0,\tfrac12,1\}$, then $Q$ is a refinement of $P$.
• If $P_1=\{0,\tfrac12,1\}$ and $P_2=\{0,\tfrac13,\tfrac23,1\}$, then $P_1\cup P_2=\{0,\tfrac13,\tfrac12,\tfrac23,1\}$ is a common refinement of both.