The author has previously shown, by the methods of Information Theory, that there is no actual limit to the small distances that can be measured, but that the cost of the measurement increases enormously when distances become really small. This strongly suggests the introduction of a probability factor in sums containing small distances. Such an attempt is made, and yields a type of summation closely related to the Cesàro sums. The method leads to the definition of a characteristic length h 2m 0c for a particle of mass m 0. For an electron this is 1 2 the Compton length, and the electromagnetec mass is of the order of 1 860 of the total mass. Divergent sums are made convergent, and convergent sums are very slightly altered. The Lamb-Retherford effect, for instance, should not be affected by this correction, which would however yield a finite electron mass. A computation of orders of magnitude shows that the conditions introduced play an important role for observations at extremely high energies.