Supports of doubly stochastic measures. (English) Zbl 0844.60002

Hestir, Kevin; Williams, Stanley C.

Bernoulli 1, No. 3, 217-243 (1995).

Summary: Recent work has shown that extreme doubly stochastic measures are supported on sets that have no axial cycles. We give a new proof of this result and examine the supporting set structure more closely. It is shown that the property of no axial cycles leads to a tree like structure which naturally partitions the support into a collection of disjoint graphs of functions from the \(x\)-axis to the \(y\)-axis and from the \(y\)-axis to the \(x\)-axis. These functions are called a limb numbering system. It is shown that if the disjoint graphs in the limb numbering system are measurable, then the supporting set supports a unique doubly stochastic measure. Further, the limb structure can be used to develop a general method for constructing sets which support a unique doubly stochastic measure.

MSC:

60A10

Probabilistic measure theory

Keywords:

extreme point; sets of uniqueness; unique doubly stochastic measure