Why "Borel" in Borel calculus?

[Ilya Zakharevich]

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