Let $F$ and $G$ be continuous univariate cdf's. For testing the hypothesis $F = G$ against general alternatives, E. Lehmann  has proposed and found certain properties of a test based on the particular measure of discrepancy $\int (F - G)^2 d\lbrack (F + G) / 2\rbrack.$ In this note will be given some additional properties of Lehmann's test (cf. also ) and a closely related test proposed by Mood .
"Properties of Some Two-Sample Tests Based on a Particular Measure of Discrepancy." Ann. Math. Statist. 27 (4) 1006 - 1016, December, 1956. https://doi.org/10.1214/aoms/1177728070