\section{Partitions}
\label{section:partitions}

\begin{definition}[Partition]
\label{definition:partition-interval}
    Let $[a, b] \subset \real$, then a \textbf{partition} of $[a, b]$ is a sequence
    \[
    P = \seqfz{x_j} = [a = x_0 \le \cdots \le x_n = b]
    \]

    The collection $\scp([a, b])$ is the set of all partitions of $[a, b]$.
\end{definition}

\begin{definition}[Tagged Partition]
\label{definition:tagged-partition}
    Let $[a, b] \subset \real$, then a \textbf{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 $[a, b]$.
\end{definition}

\begin{definition}[Mesh]
\label{definition:mesh}
    Let $P$ be a partition of $[a, b] \subset \real$, then
    \[
    \sigma(P) = \max_{1 \le j \le n}(x_j - x_{j - 1})
    \]

    is the \textbf{mesh} of $P$.
\end{definition}

\begin{definition}[Fine]
\label{definition:partition-refinement}
    Let $P = \seqfz[m]{x_j}, Q = \seqfz{y_j} \in \scp([a, b])$, then $Q$ is \textbf{finer} than $P$ if for every $0 \le j \le m$, there exists $0 \le k \le n$ 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
    \begin{enumerate}
        \item $\scp([a, b])$/$\scp_t([a, b])$ equipped with $\le$ is a upward-directed set.
        \item If $P \le Q$, then $\sigma(P) \ge \sigma(Q)$.
        \item For any $\eps > 0$, there exists $P \in \scp([a, b])$ with $\sigma(P) < \eps$.
    \end{enumerate}

    If $(P, c), (Q, d) \in \scp_t([a, b])$, then $(Q, d)$ is finer than $(P, c)$ if $Q$ is finer than $P$.
\end{definition}