In conventional system identification and state estimation problems, it is commonly assumed that the output signal of a dynamical system is sampled at every regular time interval. This paper addresses the identification and estimation problems under the Lebesgue sampling, which is a type of event-triggered sampling such that the output signal is sampled only when it crosses a specific threshold. In this paper, it is assumed that the output signal is sampled under the Lebesgue sampling rule. Then, the time interval between two samples possesses information such that the signal crosses none of the thresholds during the interval. The inter-sample information plays a key role to improve the accuracy of modeling and estimation. The problems utilizing the information are formulated. We propose likelihood-based methods of both system identification and state estimation to solve the problems. The effectiveness of the methods are illustrated in numerical examples.