Sheng Li (gmachine1729) wrote,
Sheng Li
gmachine1729

Using the Riemann integral to illustrate convergence of a net

Originally published at 狗和留美者不得入内. You can comment here or there.

Definition 1 Take an interval [公式] . A partition of it is a finite sequence of real numbers [公式] .

Definition 2 A partition [公式] is a refinement of a partition [公式] if it contains all the points of [公式] . Given any two partitions [公式] , 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 [公式] is equal to

[公式] 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. [公式]

Definition 4 A tagged partition of [公式] , consists of [公式] , and [公式] for [公式] . 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 [公式] .

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 [公式] corresponding to a tagged partition [公式] of [公式] , [公式] , [公式] , is given by

[公式]

Proposition 3 The collection of Riemann sums of a function [公式] is a net.

Proof: Proposition 2 tells us that the collection of tagged partitions of [公式] is a directed set, and the Riemann sum of [公式] is a function from this directed set to the reals. [公式]

Definition 6 We say that a function [公式] is Riemann integrable if the net of Riemann sums associated with it converges to some real value [公式] . Equivalently, for all [公式] , there exists a tagged partition such that for any refinement of it, the distance between the Riemann sum and [公式] is less than [公式] .

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