Errata for Measure Theory: A First Course

p.v. Replace “You” by “you” p.viii, line -10. Replace “corection” by “correction” p.3, line 12. Replace {X, X } by (X, X) p.3, line -8. Replace “once” by “since” p.3, line -3. Replace {ℕ, N } by (ℕ, N) p.4, line -14. Replace {X, X } by (X, X) p.6, lines 1 and 3. Replace X^{ℝ} by ℝ^{X} p.12, line 4. Delete α > p.15, line 1. Replace “spaces” by “space” p.15, line -10. Replace “is separable if it has a countable base.” by

“is separable if it has countable dense subset.”

p.15, line -3. Replace “coincide” by “coincides” p.23, line 5. Replace “complement on N” by “complement of N” p.23, line -8. Insert “a” between “is” and “nonnegative” p.31, line 6. Replace λ: ℜ → ℝ by λ: ℜ → \overline{ℝ} p.31, line 17. Replace X by ℜ ; μ by λ; and X by ℝ p.31, last line. Replace λ: ℜ → ℝ by λ: ℜ → \overline{ℝ} p.33, line -10. Insert “in” twice after “either” and “or” p.33, line -9. Insert “for” before “all” p.35, line -3. Replace “if and only if” by “if” p.35, line -2. Replace “any” by “some” p.40, lines 4 and 8. Replace twice μ( ⋃_{n} E_{n}) by λ( ⋃_{n} E_{n}) p.40, lines 17 and 19. Replace three times dμ by dλ p.42, line 6. Replace lim_{n} f by lim_{n} f_{n} p.42, line 9. Replace μ(E) = 0 by nμ(E) = 0 p.46, line 5. Delete “Finally,”

Insert at the end “(if the integrals are finite)”

p.50, line 15. Replace {x ∈ X: f ≥ 0} by {x ∈ X: f(x) ≥ 0} p.50, line 15. Replace {x ∈ X: f ≤ 0} by {x ∈ X: f(x) ≤ 0} p.50, line 24. Replace sup_{x∈X} by sup_{x∈X} p.52, line 13. Delete comma. Replace M(X, X ,) by M(X, X) p.53, line 12 (displayed formula). Replace the first f^{−} by f^{+} p.53, line 12 (displayed formula). Insert dμ after the first integral p.56, line 8 (item (a) in Problem 4.4). Replace f ≤ g with 0 ≤ f ≤ g p.56, line 11. Replace “(a) Verify that f^{+} ≤ g^{+} and g^{−} ≤ f^{−} μ-a.e., and apply” by

“(a) Apply”

p.57, line 2. Delete “a” (...and, in this case,) p.57, line -2. Insert “measurable” between “countable” and “partition” p.62, line 5. Replace “(i) and (ii)” by “(i), (ii), and (iii)” p.63, line 5. Replace “is” by “be” and replace“an” by “a” p.68, line 15 (displayed formula). Replace ≤ by = p.69, line 4. Replace “Problems” by “Problem” p.69, line 8 (displayed formula). Replace |f_{n_{k}}| = ∑_{j=k}^{∞} (|f_{n_{j+1}}| − |f_{n_{j}}|) by

|f_{n_{k}}| ≤ | ∑_{j=k}^{∞} (|f_{n_{j}}| − |f_{n_{j+1}}|)|

p.69, line 9. Replace |g| by g p.69, line -5 (displayed formula). Replace ≤ by < p.70, line 6 (displayed formula). Replace |f_{n}(x) − f_{n}(x)| by |f_{n}(x) − f_{n}(y)| p.71, line 13 Replace. “the previous problem” by “Problem 5.1” p.74, line 4 (displayed formula). Replace ∫_{[0,1]} f(x) by ( ∫_{[0,1]} |f(x)|^{p} dx)^{\frac{1}{p}} p.74, line 11 (displayed formula). Replace 2nt by 2nx p.74, line 13. Replace “Verify the” by “Verify that” p.78, line -6. Insert “of” (a sequence of functions...) p.79, line -2. Replace 1^{−n} by (−1)^{n} p.80, line 12. Replace |f′−f′′|≤ |f_{n}−f′′| for all n by |f′−f′′|≤ lim sup_{n} |f_{n}−f′′| p.80, line 14 (displayed formula). Replace ≤ ∫ |f_{n} − f′′|^{p} dμ = ∥f_{n} − f′′∥_{p}^{p} → 0 by

≤ lim sup_{n} ∫ |f_{n} − f′′|^{p} dμ = lim_{n} ∥f_{n} − f′′∥_{p}^{p} = 0

p.80, line -8. Insert “in” (i.e., it is in [f]) p.80, line -2. Insert “a” (let f be a real-valued...) p.81, line 8 (displayed formula). Replace 2g^{p} by (2g)^{p} p.82, line 15 (displayed formula). Replace ∥f_{n} − f∥_{p} by ∥f_{n} − f∥_{p}^{p} p.85, line 15. Delete “a” (which defines the real-valued...) p.85, lines -7 and -5. Replace twice E′_{k} by E_{k′} p.86, line 9. Insert “a” (let f be a real-valued...) p.87, line -9. Replace 1^{−n} by (−1)^{n} p.88, line 15. Delete “a” (This defines the real-valued...) p.91, line 18. Replace n ≥ 2m by n = 2m; and (n − m)m ≥ m^{2} by \frac{n−m}{n} = \frac{1}{2} p.91, last line. Replace “once” by “ since” p.92, line 20. Replace μ by λ p.92, line -8 (displayed formula). Replace x ∈ [0, 1 ] by x ∈ (0, 1] p.103, lines 4 and 9 (displayed formulas). Replace twice (\frac{1}{2})^{m} by 2(\frac{1}{2})^{m} p.110, line -4. Replace X^{ℝ} by ℝ^{X} p.112, line 11. Replace λ ≪ μ by ν ≪ μ p.116, line -7. Replace “once” by “since” p.117, lines 17 and 19. Replace twice E ∪ E_{n} by E ∩ E_{n} p.117, line 18. Replace “Proposition 2.2(b)” by “Proposition 2.2(a)” p.119, line 11. Replace twice F_{n} by F_{i} p.121, line 16. Replace A ⊆ A^{∗} by A ⊆ A^{∗} p.121, line -10. Replace A^{∗} by A^{∗} (A over A^{∗}) p.125, line -3 (displayed formula). Replace a by α p.126, line -5. Replace “infinite sum (which is finite)” with “indexed sum” p.127, line 5. Delete “and” p.127, line -13 (displayed formula). Replace the first ≤ by = p.128, line 14 (displayed formula). Replace ℓ^{∗}(F) by ℓ^{∗}(S) p.130, line 17. Replace “Thus.” by “Thus,” — replace dot with comma p.129, line -5. Replace (ℝ, ℑ^{∗}, μ^{∗}) by (ℝ, ℑ^{∗}, λ^{∗}) p.133, line 10 (Problem 8.3(b)). Delete “the” p.133, last line. Replace “once” by “since” p.135, line 12. Replace “refereed” by “referred” p.135, line -5 (displayed formula). Replace S\G by G\S p.135, line -3. Replace inf_{U∈ℝ} by inf_{U∈ℜ} p.136, line 1. Replace sup_{F ∈ℝ} by sup_{F ∈ℜ} p.136, line 9. Replace E by E′ p.136, line 11. Replace ⊆ by ⊇ p.136, line 14. Replace λ^{∗}(E\F_{ε}) = ε by λ^{∗}(E\F_{ε}) ≤ ε p.136, line 14. Replace = λ^{∗}(F_{ε}) + ε by ≤ λ^{∗}(F_{ε}) + ε p.143, line -4. Replace “infinite sum (which is finite)” with “indexed sum” p.145, line 15. Replace “and” by “the” p.146, line 19. Replac x ∈ X by x ∈ X p.146, line -7. Replace “a important” by “an important” p.153, line -4. Replace ∫_{X\N} f_{h^{−}} dμ −∫_{X\N} f_{h^{−}} dμ by

∫_{X\N} f_{h^{+}} dμ −∫_{X\N} f_{h^{−}} dμ

p.157, line -10. Replace λ×ν by λ×μ p.158, line -5. Replace n ∈ ℕ by m ∈ ℕ p.163, line 1. Replace “absolute” by “absolutely” — change order (alphabetical) p.164, line 21. Replace “2” by “3” (De Morgan Laws, 2)

May 25, 2015