This paper introduces measures of information for Bayesian analysis when the support of data distribution is truncated progressively. The focus is on the lifetime distributions where the support is truncated at the current age t>=0. Notions of uncertainty and information are presented and operationalized by Shannon entropy, Kullback-Leibler information, and mutual information. Dynamic updatings of prior distribution of the parameter of lifetime distribution based on observing a survival at age t and observing a failure or the residual lifetime beyond t are presented. Dynamic measures of information provided by the data about the parameter of lifetime distribution, and dynamic predictive information are introduced. These measures are applied to two well-known lifetime models. The paper concludes with some remarks on use of generalized uncertainty and information measures, and some topics for further research.