42 lines
1.7 KiB
TeX
42 lines
1.7 KiB
TeX
\section{Partitions}
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\label{section:partitions}
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\begin{definition}[Partition]
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\label{definition:partition-interval}
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Let $[a, b] \subset \real$, then a \textbf{partition} of $[a, b]$ is a sequence
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\[
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P = \seqfz{x_j} = [a = x_0 \le \cdots \le x_n = b]
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\]
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The collection $\scp([a, b])$ is the set of all partitions of $[a, b]$.
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\end{definition}
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\begin{definition}[Tagged Partition]
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\label{definition:tagged-partition}
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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 $1 \le j \le n$.
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The collection $\scp_t([a, b])$ is the set of all tagged partitions of $[a, b]$.
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\end{definition}
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\begin{definition}[Mesh]
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\label{definition:mesh}
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Let $P$ be a partition of $[a, b] \subset \real$, then
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\[
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\sigma(P) = \max_{1 \le j \le n}(x_j - x_{j - 1})
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\]
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is the \textbf{mesh} of $P$.
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\end{definition}
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\begin{definition}[Fine]
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\label{definition:partition-refinement}
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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 $0 \le k \le m$ 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
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\begin{enumerate}
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\item $\scp([a, b])$/$\scp_t([a, b])$ equipped with $\le$ is a upward-directed set.
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\item If $P \le Q$, then $\sigma(P) \ge \sigma(Q)$.
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\item For any $\eps > 0$, there exists $P \in \scp([a, b])$ with $\sigma(P) < \eps$.
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\end{enumerate}
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If $(P, c), (Q, d) \in \scp_t([a, b])$, then $(Q, d)$ is finer than $(P, c)$ if $Q$ is finer than $P$.
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\end{definition}
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