Summarizing the whole support of a random variable into minimum volume sets of its probability density function is studied in the paper. We prove that the level sets of a probability density function correspond to minimum volume sets and also determine the conditions for which the inverse proposition is verified. The distribution function of the level cuts of a density function is also introduced. It provides a different visualization of the distribution of the probability for the random variable in question. It is also very useful to prove the above proposition. The volume of the minimum volume sets varies according to its probability alpha: smaller volume implies lower probability and vice versa and larger volume implies larger probability and vice versa. In this context, 1 - alpha is the error of an erroneously classification of a new observation inside of the minimum volume set or corresponding level set. To decide the volume and/or the error of the level set that will serve to summarize the support of the random variable, a alpha - lambda plot is defined. We also study the relation of the minimum volume set approach with random set theory when cc is a random variable and extend the most specific probability-possibility transformation proposed in [System Theory, Knowledge Engineering and Problem Solving, in: Fuzzy Logic: State of the Art, vol. 12, Kluwer, 1993, pp. 103-112] to continuous piece-wise strictly monotone probability density functions.