Self-organization in the SOM and Lebesque continuity of the input distribution

Given a one dimensional SOM with a monotonically decreasing neighborhood and an input distribution which can be Lebesque continuous or not, a set of sufficient conditions and a theorem are stated which ensure probability one organization of the neuron weights. The implication of the theorem in the case of an input distribution not Lebesque continuous is a rule for choosing the number of neurons and width of the neighborhood to improve the chances of reaching an organized state in a practical implementation of the SOM. In the case of a Lebesque continuous input, self-organization in the standard SOM is proved without modifying the winner definition. Possibilities of extending the analysis to the multi-dimensional case and to a decreasing gain function are discussed