1. The problem statement, all variables and given/known data Are Lebesgue measures considered metrics? 3. The attempt at a solution I have an elementary understanding of metrics and am currently learning about Lebesgue Integration. A Lebesgue measure as a length in Euclidean space should be a metric, but what about areas and volumes? I'm thinking no because metrics are distances, which to me means length. Is there a way I can prove this? Are there necessary properties of each that agree or disagree?