Extreme Measurable Selections

Random Measurable Selections

We make the first steps towards showing a general “randomness for free” theorem for stochastic automata. The goal of such theorems is to replace randomized schedulers by averages of pure schedulers. Here, we explore the case of measurable multifunctions and their measurable selections. This involves constructing probability measures on the measurable space of measurable selections of a given me...

Bore1 measurable selections of Paretian utility functions

We show that there is a Bore1 measurable selection of Paretian utilities in ‘markets with a continuum of traders’. Theorem: Let T and X be Polish spaces, R a Bore1 subset of T x X xX, and B={(t,x): (t,x,x)ER}. Suppose that for each t, R,={(x,y): (t,x,y)ER} is a preference order on B, = {x: (t,x) EB). Then there is a Bore1 measurable function f: B-$0, l] such that for all t E IT; f,: B,+[O, I] i...

Approximations of upper and lower probabilities by measurable selections

A random set can be regarded as the result of the imprecise observation of a random variable. Following this interpretation, we study to which extent the upper and lower probabilities induced by the random set keep all the information about the values of the probability distribution of the random variable. We link this problem to the existence of selectors of a multi-valued mapping and with the...

Selections

It was not until the sixteenth century that dead bodies were submitted to medico-legal investigation. The horror which human dissections inspired in the ancients, and the religious awe with which they beheld the mortal wreck of their fellows, not only retarded the progress of anatomy and physiology, but excluded the light of medicine from the researches of the forum. The Jewish law, indeed, con...

Measurable Utility and the Measurable Choice Theorem