Partitions and Riemann Sums

Definition A partition of a closed interval $[a, b]$ with $a < b$ is a finite set of points

It splits $[a, b]$ into the $n$ subintervals $[x_{i-1}, x_i]$ with widths $\Delta x_i = x_i - x_{i-1} > 0$, and these widths add up to $b - a$.

Definition The mesh, also called the norm, of a partition $P$ is the width of its widest subinterval,

Definition A tagged partition of $[a, b]$ is a partition $P = \{x_0, \ldots, x_n\}$ together with a choice of sample points, or tags, $x_i^*$ with

Each tag may be any point of its subinterval, endpoints included. Left endpoints $x_i^* = x_{i-1}$, right endpoints $x_i^* = x_i$ and midpoints $x_i^* = \frac{x_{i-1} + x_i}{2}$ are common choices, but none is required.

Definition Let $f$ be defined on $[a, b]$ and let $P$ be a partition with tags $x_i^*$. The Riemann sum of $f$ for this tagged partition is

No sign condition on $f$ is needed. Terms where $f(x_i^*) < 0$ are negative.

Definition Let $f$ be bounded on $[a, b]$ and let $P$ be a partition. On each subinterval put

The lower and upper Darboux sums are

For every choice of tags, $m_i \le f(x_i^*) \le M_i$, so every Riemann sum for $P$ lies between $L(f, P)$ and $U(f, P)$. When $f$ is continuous, $m_i$ and $M_i$ are the minimum and maximum of $f$ on the piece.

Definition A partition $P'$ of $[a, b]$ is a refinement of $P$ if $P \subseteq P'$. For bounded $f$,

Consequently, for any two partitions $P$ and $Q$, $L(f, P) \le U(f, Q)$, which follows by comparing both with the common refinement $P \cup Q$.

Definition Suppose $f$ is bounded on $[a, b]$ and there is a sequence of partitions $P_k$ with $U(f, P_k) - L(f, P_k) \to 0$ and both sums tending to a number $I$. Then the Riemann sums on $P_k$ tend to $I$ as well, however the tags are chosen, and the choice may change from one $k$ to the next. The reason is that each such sum lies between $L(f, P_k)$ and $U(f, P_k)$, so the squeeze theorem applies.