Lions' representation theorem and applications

A Representation Theorem and Applications

We introduce a set of transformations on the set of all probability distributions over a finite state space, and show that these transformations are the only ones that preserve certain elementary probabilistic relationships. This result provides a new perspective on a variety of probabilistic inference problems in which invariance considerations play a role. Two particular applications we consi...

A Stone’s Representation Theorem and Some Applications

In this paper, we prove the following form of Stone’s representation theorem: Let ∑ be a σ -algebra of subsets of a set X . Then there exists a totally disconnected compact Hausdorff space K for which ( ∑ ,∪,∩) and (C(K),∪,∩) , where C(K) denotes the set of all clopen subsets of K , are isomorphic as Boolean algebras. Furthermore, by defining appropriate joins and meets of countable families in...

Fan-KKM Theorem in Minimal Vector Spaces and its Applications

In this paper, after reviewing some results in minimal space, some new results in this setting are given. We prove a generalized form of the Fan-KKM typetheorem in minimal vector spaces. As some applications, the open type of matching theorem and generalized form of the classical KKM theorem in minimal vector spaces are given.

Representation Theorem for Stacks

In this paper i is a natural number and x is a set. Let A be a set and let s1, s2 be finite sequences of elements of A. Then s1s2 is an element of A∗. Let A be a set, let i be a natural number, and let s be a finite sequence of elements of A. Then s i is an element of A∗. The following two propositions are true: (1) ∅ i = ∅. (2) Let D be a non empty set and s be a finite sequence of elements of...

The Riesz representation theorem