 Using the machinery of measure theory, this can be extended to functions which are only measurable (see Borel functional calculus).In that context, if "E" ⊂ σ("T") is a Borel set and "E"("x") is the characteristic function of "E", the projection operator "E"("T") is a refinement of "ei"("T") discussed above.The Borel functional calculus extends to unbounded self-adjoint operators on a Hilbert space.