  Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the
C*-algebra of bounded operators on $H.$ Suppose that $T_1,T_2,..., T_n$ are
self-adjoint operators in $B(H).$ We show that, if commutators $[T_i, T_j]$ are
sufficiently small in norm, then ``Approximately Macroscopically Unique" states
always exist for any values in a synthetic spectrum of the $n$-tuple of
self-adjoint operators. This is achieved under the circumstance for which the
$n$-tuple may not be approximated by commuting ones. This answers a question
proposed by David Mumford for measurements in quantum theory. If commutators
are not small in norm but small modulo compact operators, then ``Approximate
Macroscopic Uniqueness" states also exist.
