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*).