Consists of two separate but closely related parts. Originally published in 1966, the first section deals with elements of integration and has been updated and corrected. The latter half details the main concepts of Lebesgue measure and uses the abstract measure space approach of the Lebesgue integral because it strikes directly at the most important results—the convergence theorems.

whereas ∫ f dλ = 0, so Hence Fatou’s Lemma 4.8 may not hold unless fn 0, even in the presence of uniform convergence. 4.0. Fatou’s Lemma has an extension to a case where the fn take on negative values. Let h be in M+(X, X), and suppose that ∫ h dµ. < + ∞. If (fn) is a sequence in M(X, X) and if –h fn, then 4.P. Why doesn’t Exercise 4.0 apply to Exercise 4.N ? 4.Q. If f ∈ M+(X,X) and then µ{x ∈X : f(x) = + ∞} = 0. [Hint: If En = {x ∈ X: f(x) n}, then 4.R. If f ∈ M +(X, X) and then

exists a δ(ε) > 0 such that E ∈ X and µ(E)< δ(ε) imply that λ(E) < ε. PROOF. If this condition is satisfied and µ(E) = 0, then X(E)< ε for all ε > 0, from which it follows that λ(E) = 0. Conversely, suppose that there exist an ε > 0 and sets En ∈ X with µ(En) < 2−n and λ(En) ε. Let , so that µ(Fn) < 2 − n + 1 and λ(Fn) ε. Since (Fn) is a decreasing sequence of measurable sets, Hence λ is not absolutely continuous with respect to µ. 8.9 RADON-NIKODÝM THEOREM. Let λ and µ be σ-finite measures

extension procedure to l and F, we generate a measure space (R,F*,l*). The a-algebra F* obtained in this construction is called the collection of Lebesgue measurable sets and the measure l* on F* is called Lebesgue measure.† Although we sometimes wish to work with (R,F*,l*), it is often more convenient to deal with the smallest σ-algebra containing F than with all of F*. It is readily seen that this smallest σ-algebra is exactly the collection of Borel sets. The restriction of Lebesgue measure

addition, m(E) < +∞, then m(F − E)= m(F) − m(E). PROOF. Since m is additive, it is immediate from the fact that F = E ∪ (F − E) and E n (F − E)= ∅ that Since m(F − E)≥ 0, we have m(F)≥ m(E). If m(E)< +∞, then we can subtract m(E) from both sides of the above equation. Q.E.D. 13.10 THEOREM. (a) If (Ek)is an increasing sequence of measurable sets, then (13.8) (b) If (Fk)is a decreasing sequence of Lebesgue measurable sets and if m(F1)< +∞, then (13.9) PROOF. (a) If m(Ek)= +∞ for some k ∈

set E ⊆Jn. 16.4 COROLLARY. A set E ⊆Jn is Lebesgue measurable if and only if (16.2) PROOF. This follows immediately from Theorem 16.3 and the fact that Jn is measurable (Theorem 13.7). For an unbounded set E ⊆ Rp, the next result is useful. 16.5 THEOREM. A set E ⊆ Rpis Lebesgue measurable if and only if the sets E ∩ Jn are measurable for each n ∈N. PROOF. If E is measurable, then the result is trivial. Conversely, if each set En:= E ∩ Jn is measurable, then it follows from the fact that ,