On Lebesgue Integral Quadrature

A new type of quadrature is developed. The Gaussian quadrature, for a given measure, finds optimal values of a function's argument (nodes) and the corresponding weights. In contrast, the Lebesgue quadrature developed in this paper, finds optimal values of function (value--nodes) and the corresponding weights. The Gaussian quadrature groups sums by function argument; it can be viewed as a $n$--point discrete measure, producing the Riemann integral. The Lebesgue quadrature groups sums by function value; it can be viewed as a $n$--point discrete distribution, producing the Lebesgue integral. Mathematically, the problem is reduced to a generalized eigenvalue problem: Lebesgue quadrature value--nodes are the eigenvalues and the corresponding weights are the square of the averaged eigenvectors. A numerical estimation of an integral as the Lebesgue integral is especially advantageous when analyzing irregular and stochastic processes. The approach separates the outcome (value--nodes) and the probability of the outcome (weight). For this reason, it is especially well--suited for the study of non--Gaussian processes. The software implementing the theory is available from the authors.