Ensemble, time, and space averages as applied to turbulent quantities are discussed in Chapter 4, and pertinent properties of the averages are obtained. Those properties, together with Reynolds decomposition, are used to derive the averaged equations of motion and the one- and two-point moment or correlation equations. The terms in the various equations are interpreted. The closure problem of the averaged equations is discussed, and possible closure schemes are considered. Those schemes usually require an input of supplemental information, unless the averaged equations are closed by calculating their terms by a numerical solution of the original unaveraged equations. The law of the wall for velocities and for temperatures, the velocity- and temperature-defect laws, and the logarithmic laws for velocities and for temperatures are derived. Various notions of randomness and their relation to turbulence are considered in the light of modern ergodic theory.