The Lebesgue’s averaging method veryfication

The work is dedicated to the research of the accuracy of the Lebesgue Averaging Method (LAM) for computing spectra of resonance radiation in solving kinetic radiation transfer equation. The method is verified for the thermal radiation transfer problem in the Earth's atmosphere. The comparison of results obtained in accordance of two variants of the LAM with the results of the precise "line-by-line" calculation has been carried out. At first variant of LAM the “pure” Lebesque integral is implemented at which the value of the absorption coefficient is taken as independent variable. At second variant of LAM the integral of Lebesque–Stieltjes is used with independent variable reduced to measure of Lebesque’s set. In order to assure purity of the experiment all the computations were performed using unique method of space-angle discretization of the kinetic equation. The high accuracy of the second variant of LAM (within 5%) has been shown with a significant cut in number of arithmetic operations in need (approximately $10^3$–$10^4$ times).