We consider a continuous-time Markov chain (CTMC) whose state space is partitioned into aggregates, and each aggregate is assigned a probability measure. A sufficient condition for defining a CTMC over the aggregates is presented as a variant of weak lumpability, which also characterizes that the measure over the original process can be recovered from that of the aggregated one. We show how the applicability of de-aggregation depends on the initial distribution. The application section is a major aspect of the article, where we illustrate that the stochastic rule-based models for biochemical reaction networks form an important area for usage of the tools developed in the paper. For the rule-based models, the construction of the aggregates and computation of the distribution over the aggregates are algorithmic. The techniques are exemplified in three case studies. ; Comment: 29 pages, 9 figures, 1 table; Ganguly and Petrov are authors with equal contribution