Appendix Mathematical Background

A.1. Non-decreasing functions

DEFINITION A.1 (Non-decreasing function).– A function image is non-decreasing if

eq

A.2. Infimum and supremum

DEFINITION A.2 (Infimum, Supremum).– Let image. Then,

eq

In particular, we have the following properties:

PROPOSITION A.1.– Let image. Then:

  1. 1) image
  2. 2) image
  3. 3) image

PROPOSITION A.2 (Inf/Sup decomposition).– Let image be two sets.

eq

A.3. Continuity

DEFINITION A.3 (Left- and right-limits).– Let and . If they exist, then we denote the right-limit and

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