The Lukacs theorem and the Olkin–Baker equation

The Olkin–Baker functional equation, except of being studied inside the theory of functional equations, is closely related to the celebrated Lukacs characterization of the gamma distribution. Its deeper understanding in the case of measurable unknown functions 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,∞)2(0,∞)2 (with respect to the Lebesgue measure on R2R2) 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 R2R2 is considered.