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The good measurement practice requires that the measurement uncertainty is estimated and provided together with the measurement result. The practice today, which is reflected in the reference standard provided by the IEC-ISO "Guide to the expression of uncertainty in measurement," adopts a statistical approach for the expression and estimation of the uncertainty, since the probability theory is the most known and used mathematical tool to deal with distributions of values. However, the probability theory is not the only tool to deal with distributions of values and is not the most suitable one when the values do not distribute in a totally random way. In this case, a more general theory, the theory of the evidence, should be considered. This paper recalls the fundamentals of the theory of the evidence and frames the random-fuzzy variables within this theory, showing how they can usefully be employed to represent the result of a measurement together with its associated uncertainty. The mathematics is defined on the random-fuzzy variables, so that the uncertainty can be processed, and simple examples are given.