  A system of Goldstone bosons - stemming from a symmetry breaking $O(N) \to
O(N-1)$ - in a finite volume at finite temperature is considered. In the
framework of dimensional regularization, the partition function is calculated
to 3 loops for 3 and 4 dimensions, where Polyakov's measure for the functional
integration is applied. Although the underlying theory is the non-linear
$\sigma $ model, the 3 loop result turns out to be renormalizable in the sense
that all the singularities can be absorbed by the couplings occuring so far. In
finite volume, this property is highly non trivial and confirms the method for
the measure. We also show that the result coincides with the one obtained using
the Faddeev- Popov measure. This is also true for the maximal generalization of
Polyakov's measure: none of the additional invariant terms that can be added
contributes to the dimensionally regularized system. Our phenomenological
Lagrangian describes e.g. 2 flavor chiral QCD as well as the classical
Heisenberg model, but there are also points of contact with the Higgs model,
superconductors etc. Moreover the finite size corrections to the susceptibility
might improve the inerpretation of Monte Carlo results on the lattice.
