We study a dissipative quantum mechanical model of a projective measurement process. The measurement device comprises an oscillator circuit plus environment, where the expectation value dynamics, in the correspondence limit, are either chaotic-like or periodic dependent upon the measured state of the quantum object - in this case a qubit. We demonstrate how the classical-like trajectories of a dissipative quantum system emerge in sympathy with the projection of the qubit state, providing a model of the measurement process. Furthermore, we verify that the anticipated Born rule holds for an ensemble average of measurements.