Nonlinear sampling and Lebesgue´s integral sums

We consider nonlinear, or "event-dependent", sampling, i.e. such that the sampling instances {t_k} depend on the function being sampled. The use of such sampling in the construction of Lebesgue\´s integral sums is noted and discussed as regards physical measurement and also possible nonlinearity of singular systems. Though the limit of the sums, i.e. Lebesgue\´s integral, is linear with regard to the function being integrated, these sums are nonlinear in the sense of the sampling. A relevant method of frequency detection not using any clock, and using the nonlinear sampling, is considered, in two different versions. The mathematics and the realization arguments essentially complete each other.