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In many applications it is necessary to digitize data, knowing only that later on some random function of the digitized data will be of interest. This problem is investigated when the data digitizers are allowed to be multidimensional, i.e., they map a -dimensional data vector into one of a set of -dimensional output vectors. It is shown that very complex distortion measures arise naturally. Results are given for the error measure defined as the squared value of the difference between the function of the digitized data and the function of the undigitized data.