Refer to Exercise 15 and on A*, define * by: * (A D M) = (A). (i)

Refer to Exercise 15 and on A*, define μ* by: μ* (A D M) = μ(A).

(i) Show that μ(A – N) = μ (A).

By Exercise 15(ii), μ9 (A È M) = μ (A – N). Therefore, by part (i), we may define μ* on A̅ by: μ* (A È M) = μ (A).

(ii) Show that μ* so defined is well defined; that is, A₁ È M₁ = A₂ È M₂, where A_i and M_i, i = 1, 2 are as in Exercise 15, then μ (A₁) = μ(A₂) .

(iii) Show that μ* is a measure on A̅ (and hence on A*, by Exercise 15(iii)).

The measure space (W, A̅, μ*) (and (W, A*, μ*)) is called completion of (W, A, μ).