Another point worth noting is that on a set D of finite measure (which any measurable subset of a probability space is), L^{N}(D) is contained in L^{N-1}(D), and so if the first moment fails to exist (non-integrable, no defined expectation) then all higher moments fail and computation of order statistics fails. Of course nature doesn’t have to be modeled by statistics, but you’d be hard pressed to out-perform simple axiomatic formulations that just assume a topolgy, continuous preference functions, and get on with it and have access to higher order moments.
Another point worth noting is that on a set D of finite measure (which any measurable subset of a probability space is), L^{N}(D) is contained in L^{N-1}(D), and so if the first moment fails to exist (non-integrable, no defined expectation) then all higher moments fail and computation of order statistics fails. Of course nature doesn’t have to be modeled by statistics, but you’d be hard pressed to out-perform simple axiomatic formulations that just assume a topolgy, continuous preference functions, and get on with it and have access to higher order moments.