Let R be a σ-ring of subsets of a set X . Let S be the σ-algebra generated by R. Recall (§3.3, Problem 8) or prove that S consists of all sets in R and all complements of sets in R.
(a) Let µ be countably additive from R into [0, ∞]. For any set C ⊂ X let µ∗(C ) := sup{µ(B): B ⊂ C, B ∈ R} (inner measure). Show that
µ∗ restricted to S is a measure, which equals µ on R.
(b) Show that the extension of µ to a measure on S is unique if and only if either S = R or µ∗(X \ A) = +∞ for all A ∈ R.