  The Olkin-Baker functional equation is closely related to the celebrated
Lukacs characterization of the gamma distribution. Its deeper understanding is
essential to settle a challenging question of multivariate extensions of the
Lukacs theorem. In this paper, first, we provide a new approach to the additive
Olkin-Baker equation which holds almost everywhere on (0,\infinity)^2 (with
respect to the Lebesgue measure on R^2) under measurability assumption. Second,
this new approach is adapted to the case when unknown functions are allowed to
be non-measurable and the complete solution is given in such a general case.
Third, the Olkin-Baker equation holding outside of a set from proper linearly
invariant ideal of subsets of R^2 is considered.
