Vorontsov and Rembovsky pointed out recently a problem that unphysical states appear after an approximate quantum measurement of the Pegg-Barnett phase [Phys. Lett. A 254 (1999), 7; 262 (1999), 486]. To give a solution to this problem, we introduce a quantum phase in the extended number-state space and show that there is no Vorontsov-Rembovsky problem in the extended space. In the infinite-dimensional limit, the present extended formalism gives the same results in almost all cases as the Pegg-Barnett formalism. Also, it is shown that unphysical states can be suppressed by using the projection operator to the physical number space.