Riemann decomposition

A Riemann decomposition is a pair of a family of support points up to and intermediate points up to , ${\displaystyle \xi _{0}}$ ${\displaystyle \xi _{n({\mathcal {R}})}}$ ${\displaystyle \alpha _{1}}$ ${\displaystyle \alpha _{n({\mathcal {R}})}}$

${\displaystyle {\mathcal {R}}=((\xi _{j})_{j=0}^{n({\mathcal {R}})};(\alpha _{j})_{j=1}^{n({\mathcal {R}})})}$

which breaks down an interval as follows: ${\displaystyle [a,b]}$

${\displaystyle a=\xi _{0}<\xi _{1}<\ldots <\xi _{n({\mathcal {R}})}=b}$ and ${\displaystyle \alpha _{j}\in [\xi _{j-1},\xi _{j}],j=1,\ldots ,n({\mathcal {R}})}$

This means that the edge points are at the same time the largest and the smallest support point, and the intermediate points can be located anywhere between the support points.
The fineness of a Riemann decomposition is defined as the maximum difference between two support points:

${\displaystyle |{\mathcal {R}}|=\max\{(\xi _{j}-\xi _{j-1}):j=1,\ldots ,n({\mathcal {R}})\}}$

The set of all Riemann decompositions of an interval is given by the relation to the directed set : ${\displaystyle \preceq }$

${\displaystyle {\mathcal {R}}_{1}\preceq {\mathcal {R}}_{2}:\Leftrightarrow |{\mathcal {R}}_{1}|\leq |{\mathcal {R}}_{2}|}$

Networks can now be defined over this directed set , for example the Riemann integral is defined over such a network.

See also

Variation (math)