I looked through several "standard books", and could not find an answer: When one considers Borel calculus of a (bounded) normal/self-adjoint operator, one can apply any function which is l-infinity w.r.t. the spectral measure of the operator [*]. So, in principle, one could restrict attention to functions which are l-infinity w.r.t. any Borel measure (call them "admissible"). In particular, any bounded Borel function is admissible. But, a priory, the class of admissible functions may be larger than the class of bounded Borel functions. Do these classes coincide? (One can ask the same about bounded admissible functions...) Likewise: consider a set U such that for any Borel measure mu one can write U = Ub UNION Un as below. Is it Borel? [*] Here I assume that the notion of "measurable" is modified in the same sense as when defining Lebesgue measure: a set U is mu-measurable if it is Ub UNION Un, with Ub Borel, and Un is a subset of a Borel set of mu-measure 0. Thanks, Ilya