On the Lp-metric between a probability distribution and its distortion

In actuarial theory, the Lp-metric is used to evaluate how well a probability distribution approximates another one. In the context of the distorted expectation hypothesis, the actuary replaces the original probability distribution by a distorted probability, so it makes sense to interpret the Lp-metric between them as a characteristic of the underlying random variable. We show in this paper that this is a characteristic of the variability of the random variable, study its properties and give some applications.