Perfect i.i.d Processes

This note proves a theorem about i.i.d. i.e. independent and indentically distributed processes, when the index space is a measure space. The statement of the problem corresponding to the theorem proved in this paper appears in [1], in which the concept of a sample distribution limit corresponds to the concept of a perfect i.i.d process in this paper.
Theorems proved in this theme, regarding existing and non-existence, have been shown in the economics literature, when the index set is [0; 1], in [2], [3], [4], [5]. The approach taken in this
paper is perhaps, surprisingly elementary. We may apply standard measure extension theorems to show existence. These may be found in [6], [7].