In this study, we prove that a countably infinite number of one-parameterized one-dimensional dynamical systems preserve the Lebesgue measure and are ergodic for the measure. The systems we consider connect the parameter region in which dynamical systems are exact and the one in which almost all orbits diverge to infinity and correspond to the critical points of the parameter in which weak chaos tends to occur (the Lyapunov exponent converging to zero). These results are a generalization of the work by Adler and Weiss. Using numerical simulation, we show that the distributions of the normalized Lyapunov exponent for these systems obey the Mittag-Leffler distribution of order 1/2.