Sobolev’s Theorem is the most fundamental theorem in the theory of Invariant Cubature Formulas (ICFs). In this paper, a quantitative structure is established for the classical ICFs. Enlightened by this structure, the author generalizes the concept of ICFs and extends the Sobolev’s Theorem to the case of generalized ICFs. Several illustrative examples are given. Sobolev’s Theorem is the most fundamental theorem in the theory of Invariant Cubature Formulas (ICFs). In this paper, a quantitative structure is established for the classical ICFs. Enlightened by this structure, the author generalizes the concept of ICFs and extends the Sobolev’s Theorem to the case of generalized ICFs. several illustrative examples are given.