There are a few articles on inequality measurement which deal with uncertainty, particularly when the ranking of cohorts may not be fixed. The authors of the present paper give a set of axioms implying such a class of inequality measures under uncertainty that is one-parameter extension of the generalized Gini mean over the distribution of average allocations. The extension consists of a quadratic term accounting for inter-personal correlations. In particular, presented measure can simultaneously accommodate a preference for ``shared destiny'', a preference for probabilistic mixtures over unfair allocations, and a preference for fairness ``for sure'' over fairness in expectation.