Mathematical Economics

We show that there exists an antimonotone relation between the open (closed) balls of fixed center and radius w.r.t. metrics that obey functional inequalities confirming the aforementioned remark concerning . 

We begun our study of metric properties with the finitary notion of boundedness. The balls can be readily used in order to define it as a natural extension of the notion of boundness on the real line (w.r.t. the usual metric). Specifically a subset of a metric space is bounded  iff it can be covered by an open (equivalently closed) ball. The center of the covering ball need not be an element of the subset at hand although we have shown that when such a ball exists the center can always be chosen to lie inside this subset.

We have shown that any finite set is (universally) bounded, while the open and closed balls are by construction bounded. The notion is hereditary. If a set is bounded then any subset is also bounded. The dual is evident. If a set is not bounded then any superset is also unbounded. Obviously the notion depends crucially on the metric. For example the discrete space is always bounded. This is in contrast with any Euclidean space (i.e. the  equipped with the usual metric) which is not bounded-the same is true when equipped with any of the metrics we have been discussing except for the discrete one, this also implies that the metric space consisting of the set of bounded real functions on X with the uniform metric is not generally bounded (i.e. universally bounded), etc. 

We have examined further details on boundness including: several (counter-) examples of boundness in function spaces using uniform boundness, some limiting variance, a characterization of boundedness for metric subspaces, and an equivalent definition that is useful for the introduction of total boundness.

Strenghtening thus, the boundedness characterization we have introduced the notion of total boundness. We have started  our examination of the latter. We have given properties, e.g. that the centers of the covering balls can be chosen inside the totally bounded set, as well as examples-counterexamples (the analytical complexity of the notion clearly manifested itself on that counter examples were easier to come up to). Given the analytical complexity of the verification of total boundness, we need the introduction of the notions of covering numbers and metric entropy in order to construct examples and further properties.

Notes on the above can be found here, here, here, here, here, and here. 

The whiteboards from analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found  here, here, and here.

Exercise: Show that a subset A of a metric space X is bounded, iff it is a bounded space when considered as metric-subspace of X.