This work analyzes the turbulent velocity distribution in proximity of the wall using the finite-scale Lyapunov theory just presented in previous works. This theory is here applied to the steady boundary layer under the hypothesis of moderate pressure gradient and fully developed flow along the streamwise direction. The analysis gives an equation for the velocities correlation and identifies the parameters of the expression of the average velocity through the statistical properties of the velocity correlation functions. In particular, the von K\'arm\'an constant, theoretically calculated, is about 0.4, and the dimensionless Prandtl's length is in function of the Taylor-scale Reynolds number. The study provides the average velocity distribution and gives also the variation laws of the other variables, such as Taylor scale and Reynolds stress. The obtained results show that the finite-scale Lyapunov theory is adequate for studying the turbulence in the proximity of the wall.