  We investigate properties of the invariant measure for the $A_0$ gauge field
in finite temperature gauge theories both on the lattice and in the continuum
theory. We have found the cancellation of the naive measure in both cases. The
result is quite general and holds at any finite temperature. We demonstrate,
however, that there is no cancellation at any temperature for the invariant
measure contribution understood as Z(N) symmetrical distribution of gauge field
configurations. The spontaneous breakdown of Z(N) global symmetry is entirely
due to the potential energy term of the gluonic interaction in the effective
potential. The effects of this measure on the effective action, mechanism of
confinement and $A_0$ condensation are discussed.
