Editing Riemann integral (section)

=== Partitions of an interval ===
{{Main|Partition of an interval}}
A '''partition''' of an interval {{math|[''a'', ''b'']}} is a finite sequence of numbers of the form
<math display="block">a = x_0 < x_1 < x_2 < \dots < x_i < \dots < x_n = b</math>

Each {{math|[''x<sub>i</sub>'', ''x''<sub>''i'' + 1</sub>]}} is called a '''sub-interval''' of the partition. The '''mesh''' or '''norm''' of a partition is defined to be the length of the longest sub-interval, that is,
<math display="block">\max \left(x_{i+1}-x_i\right), \quad i \in [0,n-1].</math>

A '''tagged partition''' {{math|''P''(''x'', ''t'')}} of an interval {{math|[''a'', ''b'']}} is a partition together with a choice of a sample point within each sub-interval: that is, numbers {{math|''t''<sub>0</sub>, ..., ''t''<sub>''n'' − 1</sub>}} with {{math|''t<sub>i</sub>'' ∈ [''x<sub>i</sub>'', ''x''<sub>''i'' + 1</sub>]}} for each {{mvar|i}}. The mesh of a tagged partition is the same as that of an ordinary partition.

Suppose that two partitions {{math|''P''(''x'', ''t'')}} and {{math|''Q''(''y'', ''s'')}} are both partitions of the interval {{math|[''a'', ''b'']}}. We say that {{math|''Q''(''y'', ''s'')}} is a '''refinement''' of {{math|''P''(''x'', ''t'')}} if for each integer {{mvar|i}}, with {{math|''i'' ∈ [0, ''n'']}}, there exists an integer {{math|''r''(''i'')}} such that {{math|''x<sub>i</sub>'' {{=}} ''y''<sub>''r''(''i'')</sub>}} and such that {{math|''t<sub>i</sub>'' {{=}} ''s<sub>j</sub>''}} for some {{mvar|j}} with {{math|''j'' ∈ [''r''(''i''), ''r''(''i'' + 1)]}}. That is, a tagged partition breaks up some of the sub-intervals and adds sample points where necessary, "refining" the accuracy of the partition.

We can turn the set of all tagged partitions into a [[directed set]] by saying that one tagged partition is greater than or equal to another if the former is a refinement of the latter.