Extension of the Generalized Lebesgue–Feynman–Smolyanov Measure on a Hilbert Space

As is well known, on an infinite-dimensional Hilbert space, there is no countably additive sigma-finite locally finite nonzero translation-invariant nonnegative Borel measure (Andre Weil’s theorem, [1]). For this reason, to formalize the Feynman path integrals [2], one has to introduce a generalized translation-invariant measure (Lebesgue–Feynman in the sense of the definition in [2]) as a linear functional on some space of functions. The present paper proposes a natural extension of one of these functionals that were introduced in [3] and called there the generalized Lebesgue measure (henceforth, we call this (generalized) measure the Lebesgue–Feynman–Smolyanov). The extension makes it possible to give a precise mathematical meaning to the Schrödinger quantization of noncylindrical Hamiltonians for Hamiltonian systems with infinitely many degrees of freedom [3]: in particular, to give a correct mathematical solution to the problem of infinite vacuum energy at the bosonic quantization of the “free” electromagnetic field (N. N. Bogoliubov, D. V. Shirkov, Introduction to the Theory of Quantized Fields, John Wiley & Sons, New York–Chichester–Brisbane, 1980); invariant measures themselves have recently been used for the mathematical description of the phenomena of quantum anomalies [4, 5, 6].