Feynman path integrals and Lebesgue-Feynman measures

Feynman path integrals (Feynman functional integrals) are defined as integrals with respect to a generalized measure, called the Lebesgue-Feynman measure. This is an infinite-dimensional analogue of the classical Lebesgue measure on finite-dimensional Euclidean space. This definition, which is a formalization of Feynman's original definition, is different from those used previously in the mathematical literature. It makes it possible to give a description of the origin of quantum anomalies which is a mathematically correct version of the description given in the book Path Integrals and Quantum Anomalies by K.~Fujikawa and H.~Suzuki (Oxford, 2004) (and wrongly described as incorrect by P.~Cartier and C.~DeWitt-Morette in their book Functional Integration: Action and Symmetries (Cambridge Univ.\ Press, Cambridge, 2006)).