  We consider the qubit initial state preparation due to the nonselective
measurements on an overcomplete basis, when the number of outcomes $N=3$. To be
specific, we have chosen the dephasing model and applied the conditions for a
post-measurement state to ensure a gradual coherence enhancement at the initial
stage of the system evolution. It is shown that contrary to the von
Neumann-L\"uders projection scheme with $N=2$, the most mixed post-measurement
state of the qubit in the general case can depend on the bath temperature. This
remarkable feature allows one to consider a ``temperature controlled'' initial
purification of the open quantum system. We also consider some kinds of the
repeated nonselective measurements from the viewpoint of an eventual system
purification and analyze the important peculiarities of such schemes.
