Complete measure

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).