Equality of energy average and ensemble average in the statistical theory of nuclear reactions

The equality of the two averages is shown for $S$ matrix elements and cross sections (and other quantities of physical interest) by expanding the scattering matrix in powers of the interaction matrix elements (Born series), and by demonstrating the equality of the two averages for each term separately of the resulting simple or double (or multiple) Born series. As an input, we use the equality of ensemble average and level average over all possible combinations of interaction matrix elements and level energies.