Computational complexity of the integration problem for anisotropic classes

We determine the exact order of epsilon-complexity of the numerical integration problem for the anisotropic class W-infinity(r)(I-d) and H-infinity(r)(I-d) with respect to the worst case randomized methods and the average case deterministic methods. We prove this result by developing a decomposition technique of Borel measure on unit cube of d-dimensional Euclidean space. Moreover by the imbedding relationship between function classes we extend our results to the classes of functions W-p(Lambda) (I-d) and H-p(Lambda) (I-d). By the way we highlight some typical results and stress the importance of some open problems related to the complexity of numerical integration.