A measurable space is a set X and a σ-ring S of subsets of X with the property that ⋃ S = X. Ordinarily it causes no confusion to denote a measurable space by the same symbol as the underlying set X; on the occasions when it is desirable to call attention to the particular σ-ring under consideration, we shall write (X,S) for X. It is customary to call a subset E of X measurable if and only if it belongs to the σ-ring S. This terminology is not meant to indicate that S is the σ-ring of all µ*-measurable sets with respect to some outer measure µ*, nor even that a non trivial measure is or may be defined on S.
KeywordsMeasurable Function Measure Space Uniform Convergence Simple Function Measure Zero
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