  The effective action of N=2 gauge multiplets in general includes
higher-dimension UV finite nonholomorphic corrections integrated with the full
N=2 superspace measure. By adding a hypermultiplet in the adjoint
representation we study the effective action of N=4 SYM. The nonanomalous SU(4)
R-symmetry of the classical N=4 theory must be also present in the on-shell
effective action, and therefore we expect to find similar nonholomorphic terms
for each of the scalars in the hypermultiplet. The N=2 path integral
quantization formalism developed in projective superspace allows us to compute
these hypermultiplet nonholomorphic terms directly in N=2 superspace. The
corresponding gauge multiplet expression can be successfully compared with the
result inferred from a N=1 calculation in the abelian subsector.
