Complete measure

From Academic Kids

In measure theory, a complete measure is a measure in which every subset of every null set is measurable (having measure 0).

Every measure has an extension that is complete. The smallest such extension is called the completion of the measure.

Suppose μ is a measure on some set X, with σ-algebra F. The completion of μ can be constructed as follows. Let N be the set of all subsets of null sets of μ, and let G be the σ-algebra generated by F and N. There is only one way to extend μ to this new σ-algebra: for every C in G, μ'(C) is defined to be the infimum of μ(D) over all D in F of which C is a subset. Then μ' is a complete measure, and is the completion of μ.

In the above construction it can be shown that every member of G is of the form A U B for some A in F and some B in N, and μ'(A U B) = μ(A).

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