A decomposable inequality measure is defined as a measure such that the total inequality of a population can be broken down into a weighted average of the inequality existing within subgroups of the population and the inequality existing between them. Thus, decomposable measures differ only by the weights given to the inequality within the subgroups of the population. It is proven that the only zero-homogeneous "income-weighted" decomposable measure is Theil's coefficient (T) and that the only zero-homogeneous "population-weighted" decomposable measure is the logarithm of the arithmetic mean over the geometric mean (L). More generally, it is proved that T and L are the only decomposable inequality measures such that the weight of the "within-components" in the total inequality of a partitioned population sum to a constant. More general decomposable measures are also analyzed.