A fundamental theorem on one-parameter continuous groups of projective functional transformations

Let Lx, Lxs, Lx1, L1s be real bounded integrable functions of the real variables x and s as indicated (a ≤ x, s ≤ b) and let us denote Riemann integration on (a, b) by the repetition of a superscript and subscript in the same term unless one of them is enclosed in a parenthesis.