Definition 1 Take an interval . A partition of it is a finite sequence of real numbers .
Definition 2 A partition is a refinement of a partition if it contains all the points of . Given any two partitions , one can always find their common refinement, which consists of all their points in increasing order.
Definition 3 The norm or mesh of the partition is equal to
Proposition 1 The collection of partitions of any closed interval is a directed set.
Proof: Under set inclusion, it is a preorder since reflexivity and transitivity are satisfied. The common refinement of any finite collection of partitions is always an upper bound of it.
Definition 4 A tagged partition of , consists of , and for . A refinement of it is the same as the refinment of a partition with the additional requirement that the set of points inside the intervals of the refinement contains the .
Proposition 2 The collection of tagged partitions of any closed interval is a directed set.
Proof: Similar as that of Proposition 1.
Definition 5 The Riemann sum of a function corresponding to a tagged partition of , , , is given by
Proposition 3 The collection of Riemann sums of a function is a net.
Proof: Proposition 2 tells us that the collection of tagged partitions of is a directed set, and the Riemann sum of is a function from this directed set to the reals.
Definition 6 We say that a function is Riemann integrable if the net of Riemann sums associated with it converges to some real value . Equivalently, for all , there exists a tagged partition such that for any refinement of it, the distance between the Riemann sum and is less than .