Complete Measure - Construction of A Complete Measure

Construction of A Complete Measure

Given a (possibly incomplete) measure space (X, Σ, μ), there is an extension (X, Σ0, μ0) of this measure space that is complete. The smallest such extension (i.e. the smallest σ-algebra Σ0) is called the completion of the measure space.

The completion can be constructed as follows:

  • let Z be the set of all subsets of μ-measure zero subsets of X (intuitively, those elements of Z that are not already in Σ are the ones preventing completeness from holding true);
  • let Σ0 be the σ-algebra generated by Σ and Z (i.e. the smallest σ-algebra that contains every element of Σ and of Z);
  • there is a unique extension μ0 of μ to Σ0 given by the infimum

Then (X, Σ0, μ0) is a complete measure space, and is the completion of (X, Σ, μ).

In the above construction it can be shown that every member of Σ0 is of the form A ∪ B for some A ∈ Σ and some B ∈ Z, and

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