Abstract The classes of the L p,∞ - and L p -metrics play an important role to develop a probability theory in fuzzy sample spaces. All of these metrics are known to be separable, but not complete. The classes are closely related as for each L p,∞ -metric there exists some L p -metric which induces the same topology. This paper deals with the completion of the L p,∞ - and L p -metrics. We can also show that the relationship between the classes of L p,∞ - and L p -metrics still holds for the obtained respective classes of their completions.