5.1 Partition

Definition 5.1.1 (Partition).label Let $[a,b]\suf\R$, then a partition of $[a,b]$ is a sequence

The collection $\scr{P}([a,b])$ is the set all partitions of $[a,b]$

Definition 5.1.2 (Tagged Partition).label Let $[a,b]\suf\R$ then a tagged partition of $[a,b]$ is a pair $(P=\curl{x_j}^{n}_{j=0}, c=\curl{c_j}^{n}_{j=0})$ such that $c_{j}\in [x_{j-1},x_{j}]$ for each $1\leq j\leq n$. The collection $\scr{P}_{t}([a,b])$ is the set of all tagged parititions of $[a,b]$.

Definition 5.1.3 (Mesh).label Let $P$ be a partition of $[a,b]\suf\R$ then $\s(P)\define \MAX{1\leq j\leq n}(x_{j}-x_{j-1})$ is the mesh of $P$.

Definition 5.1.4 (Fine).label Let $P=\curl{x_j}^{n}_{j=0},Q=\curl{y_j}^{n}_{j=0}\in\scr{P}([a,b])$, then $Q$ is finer than $P$, denoted $P\leq Q$, if $\fall 0\leq j\leq m,\exists 0\leq k\leq n$ such that $x_{j}=y_{k}$. For any $P\leq Q\in \scr{P}([a,b])$ then

1. (1)

   $\scr{P}([a,b])$ or $\scr{P}_{t}([a,b])$ is equipped with $\leq$ is a upward-directed set

2. (2)

   If $P\leq Q$, then $\s(P)\geq \s(Q)$

3. (3)

   $\fall \e>0,\exists P\in\scr{P}([a,b])$ with $\s(P)<\e$