Setup

Let $f[a,b] \rightarrow$ R be a bounded real function. Partition the interval [a,b] into \begin{equation} a = a_0 < a_1 < … < a_n = b \end{equation}

Call such a partition P. For each subinterval $[x_{i-1}, x_i]$ of this P,

Define the lower and upper Riemman sum as follows:

\begin{equation} L(P, f) = \sum_{i=1}^n m_i*(x_i-x_{i-1}), \ U(P,f) = \sum_{i=1}^n M_i*(x_i-x_{i-1}) \ \end{equation}

where $m_i = \{ \inf f(x) : x \in [x_{i-1}, x_i] \}$, $M_i = \{\sup f(x) : x \in [x_{i-1}, x_i]\}$

Refinement of partition

Definition 7.2.2 (As in reference) A partition Q is a refinement of a partition P if Q contains all of the points of P; that is $ P \subseteq Q $.

References:

Abbott, S. (2015). Understanding Analysis. Springer New York. https://doi.org/10.1007/978-1-4939-2712-8