We provide the basis for a rigorous construction of the Schwinger func
tions of the pure SU(2) Yang-Mills field theory in four dimensions (in
the trivial topological sector) with a fixed infrared cutoff but no u
ltraviolet cutoff, in a regularized axial gauge. The construction expl
oits the positivity of the axial gauge at large field. For small field
s, a different gauge, more suited to perturbative computations is used
; this gauge and the corresponding propagator depends on large backgro
und fields of lower momenta. The crucial point is to control (in a non
-perturbative way) the combined effect of the functional integrals ove
r small field regions associated to a large background field and of th
e counterterms which restore the gauge invariance broken by the cutoff
. We prove that this combined effect is stabilizing if we use cutoffs
of a certain type in momentum space. We check the validity of the cons
truction by showing that Slavnov identities (which express infinitesim
al gauge invariance) do hold non-perturbatively.