Topology and the Construction of Extreme Quasi-Measures

Simple topological measures and a lifting problem

“…One may use the Mayer-Vietoris sequence to show that the property that disjoint unions of co-connected sets stay co-connected holds if theČech cohomology group H 1 (X, Z) vanishes. For reasonable spaces the converse is also true (see [9]). On spaces of Aarnes genus zero, one can construct topological measures quite freely, and it is proved in [9] that this construction yields arbitrarily good approximations to any simple topological measure.…”

“…For reasonable spaces the converse is also true (see [9]). On spaces of Aarnes genus zero, one can construct topological measures quite freely, and it is proved in [9] that this construction yields arbitrarily good approximations to any simple topological measure. An amazing result, proved by Butler in [4], is that on q-spaces which are CW -complexes of dimension at least 2, arbitrary topological measures can be approximated by extreme topological measures.…”

“…Intuitively this means the absence of holes. In [9] it was proved that a simple topological measure is completely determined by its restriction to the family of closed solid subsets. Later this was proved by Aarnes for general topological measures [2].…”

“…Simple measures are extreme, but there are non-simple extreme measures (see [2]). On a torus and on generalized tori it is not too hard to construct some topological measures (see [6] in the case of general topological measures, and [9] and [10] in the case of simple topological measures). On the torus some simple topological measures kill all contractible sets.…”

“…We will say that F is liftable to Y , or p-liftable, if there is a strongly linked family G of connected subsets of X which lifts F. Note that if F is liftable to the universal covering ( 1 ) Some authors call strongly linked families simply linked families, but I feel that the term linked only suggests connectedness of the graph. In [9] strongly linked families are called PNI-families, which is short for pairwise non-empty intersection.…”