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