The Ball Sigma-Field and Measurability of Suprema

The ball σ-field on D is the smallest σ-field containing all the open (and/or closed) balls in D. In general, this is smaller than the Borel σ-field, although the two σ-fields are equal for separable spaces (Problems 1.7.3 and 1.7.4). For some nonseparable spaces, it is even fairly common that maps are ball measurable even though they are not Borel measurable. Thus one may wonder about the possibility of a weak convergence theory for ball measurable maps. It turns out that the set of ball measurable f ∈ C _b (D) is rich enough to make this fruitful, but at the same time it is so rich that the theory is a special case of the theory that we have discussed so far.