11.1 Partitions
Definition 11.1.1 (Partition). Let $[a, b] \subset \real$, then a partition of $[a, b]$ is a sequence
The collection $\scp([a, b])$ is the set of all partitions of $[a, b]$.
Definition 11.1.2 (Tagged Partition). Let $[a, b] \subset \real$, then a tagged partition of $[a, b]$ is a pair $(P = \seqfz{x_j}, c = \seqf{c_j})$ such that $c_{j} \in [x_{j - 1}, x_{j}]$ for each $1 \le j \le n$.
The collection $\scp_{t}([a, b])$ is the set of all tagged partitions of $[a, b]$.
Definition 11.1.3 (Mesh). Let $P$ be a partition of $[a, b] \subset \real$, then
is the mesh of $P$.
Definition 11.1.4 (Fine). Let $P = \seqfz[m]{x_j}, Q = \seqfz{y_j}\in \scp([a, b])$, then $Q$ is finer than $P$ if for every $0 \le j \le m$, there exists $0 \le k \le m$ such that $x_{j} = y_{k}$. For any $P, Q \in \scp([a, b])$, denote $P \le Q$ if $Q$ is finer than $P$, then
$\scp([a, b])$/$\scp_{t}([a, b])$ equipped with $\le$ is a upward-directed set.
If $P \le Q$, then $\sigma(P) \ge \sigma(Q)$.
For any $\eps > 0$, there exists $P \in \scp([a, b])$ with $\sigma(P) < \eps$.
If $(P, c), (Q, d) \in \scp_{t}([a, b])$, then $(Q, d)$ is finer than $(P, c)$ if $Q$ is finer than $P$.