Classical mechanics is presented here in a unary operator form, constructed using the binary multiplication and Poisson bracket operations that are given in a phase space formalism, then a Gibbs equilibrium state over this unary operator algebra is introduced, which allows the construction of a Hilbert space as a representation space of a Heisenberg algebra, giving an operator algebraic variant of the Koopman--von Neumann approach. A classically natural approach to the Poisson bracket in a Koopman--von Neumann Hilbert space formalism gives a classical logic that is the same as quantum logic: in both cases the logic of projections on Hilbert space. In this form, the measurement theory for unary classical mechanics can be the same as and inform that for quantum mechanics, expanding classical mechanics to include noncommutative operators so that it is close to quantum mechanics, instead of attempting to squeeze quantum mechanics into a classical mechanics mold. The measurement problem as it appears in unary classical mechanics suggests a classical approach that can also be successfully applied to the measurement problem of quantum mechanics.