If X and Y are any two sets (not necessarily subsets of the same space), the Cartesian product X × Y is the set of all ordered pairs (x,y), where x ε X and y ε Y. The best known example of a Cartesian product is the Euclidean plane, which is most often viewed as the product of two coordinate axes. Most of the development in the sequel uses the words and concepts suggested by this example. Thus, for instance, if A ⊂ X and B ⊂ Y, we shall call the set E = A × B (a subset of X × Y) a rectangle and we shall refer to the component sets A and B as its sides. (Observe that our usage here differs from the classical terminology which speaks of rectangles only if the sides are intervals.)
KeywordsLebesgue Measure Measure Space Disjoint Union Product Space Product Measure
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