Why "Borel" in Borel calculus?

Discussion in 'Math Research' started by Ilya Zakharevich, Sep 27, 2008.

  1. 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.

    Ilya Zakharevich, Sep 27, 2008
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