Why saturated probability spaces are necessary

An atomless probability space (Ω,A,P) is said to have the saturation property for a probability measure μ on a product of Polish spaces X×Y if for every random element f of X whose law is margX(μ), there is a random element g of Y such that the law of (f,g) is μ. (Ω,A,P) is said to be saturated if it has the saturation property for every such μ. We show each of a number of desirable properties holds for every saturated probability space and fails for every non-saturated probability space. These include distributional properties of correspondences, such as convexity, closedness, compactness and preservation of upper semi-continuity, and the existence of pure strategy equilibria in games with many players. We also show that any probability space which has the saturation property for just one “good enough” measure, or which satisfies just one “good enough” instance of the desirable properties, must already be saturated. Our underlying themes are: (1) There are many desirable properties that hold for all saturated probability spaces but fail everywhere else; (2) Any probability space that out-performs the Lebesgue unit interval in almost any way at all is already saturated.