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.