Measure, Integration & Real Analysis

Sheldon Axler

errata

I tried hard to make Measure, Integration & Real Analysis error-free, but spotting mistakes in one's own writing can be difficult in a book-length work. All typos in the print version that I know about are listed below. These typos have all been corrected in the current electronic version dated 4 May 2022 (freely available here). The electronic version of the book available from Springer and on several other websites is older and does not include corrections to the typos below.

• page 4, line 5: “set” should be “nonempty set”. (corrected 9 July 2020)

• page 6, line 7 from bottom: “first equality” should be “first line”. (corrected 24 April 2022)

• page 18, line 2 of second bullet point of 2.11: “[2, ∞]” should be “[2, ∞)”. (corrected 3 September 2020)

• page 23, lines 8 and 10: “D” should be “V ”. (corrected 1 June 2020)

• page 40, Exercise 24: This exercise is correct as stated, but its solution requires tools beyond the scope of this book. Thus the hypotheses have been slightly changed in the electronic version of the book (freely available here) to make this exercise solvable with the tools presented in the book. (corrected 10 August 2021)

• page 47, line 3: “A, B ∈ R” should be “A, B ⊂ R”. (corrected 22 September 2020)

• page 65, first line of 2.89: “measure space” should be “measurable space”. (corrected 1 June 2020)

• page 65, displayed equation defining f_k (x): “−k < f (x)” should be “−k ≤ f (x)” and “f (x) ≤ −k” should be “f (x) < −k”. (corrected 25 August 2020)

• page 66, line 10 from bottom: “=” should be “⊂ ”. (corrected 7 October 2020)

• page 79, line 3 from bottom: “χ_Bk” should be “B_k”. (corrected 8 April 2022)

• page 85, Exercise 8: “Given” should be “Give”. (corrected 21 July 2020)

• page 91, lines 2 and 9 from bottom: “inf f ” should be “inf g”. (corrected 21 July 2020)

• page 115, Exercise 6: “h = 0” should be “h(s) = 0 for almost every s ∈ R”. (corrected 8 August 2020)

• page 135, Exercise 4: “if and only if ” should be “if ”. (corrected 11 August 2021)

• page 139, line 13 from bottom: “integrated integrals” should be “iterated integrals”. (corrected 8 April 2022)

• page 168, first line of proof of 6.46: “then” should be “Then”. (corrected 26 January 2021)

• page 226, line 10 from bottom: “existence proof” should be “proof”. (corrected 17 April 2022)

• page 236, Exercise 25, part (b): Replace the hypothesis from part (a) that “1 ≤ p ≤ 2” by “1 < p ≤ 2”. (corrected 22 December 2021)

• page 251, Exercise 7: “space” should be “space V and f ∈ V ”. “converges” (twice) should be “equals f ”. “injective” should be “injective and surjective”. (corrected 22 March 2021)

• page 252, Exercise 11: Add the hypothesis that Γ is a countable set. (corrected 24 March 2021)

• page 274, line 10 from bottom: “Refine” should be “Redefine”. (corrected 21 August 2020)

• page 284, line 3 from bottom: “Tg” should be “Tf ”. (corrected 1 June 2020)

• page 286, two lines above 10.16: “closed” should be “open”. (corrected 17 April 2022)

• page 290, line 5 from bottom: “such” should be “such that”. (corrected 6 April 2021)

• page 311, Exercise 24: “h = χ_E” should be “|h| = χ_E”. (corrected 21 November 2021)

• page 313, last line of text box at top of page: “nor V ” should be “nor B(V) ”. (corrected 1 February 2021)

• page 321, line 11 from bottom: “T - αf ” should be “T - αI ”. (corrected 1 June 2020)

• page 328, line 5 from bottom: “K” should be “Γ”. (corrected 17 April 2022)

• page 338, Exercise 21: “operator” should be “operator on a complex Hilbert space”. (corrected 14 November 2021)

• page 340, line 4 from bottom: “(π, π]” should be “(-π, π]”. (corrected 26 April 2021)

• page 348, Definition 11.19: Each “f ” should be “u”. (corrected 1 May 2021)

• page 350, second paragraph: “(-π, π]” (twice) should be “R”. (corrected 17 April 2022)

• page 351, line 4 from bottom: “each” should be “each z”. (corrected 30 April 2021)

• page 362, Exercise 11: A factor of 1/(2πi) is missing from right side of displayed equation. (corrected 1 June 2020)

• page 362: The last line of Exercise 13 should be “that if f ∈ L¹(∂D), then f ∗ g ≠ g.” (corrected 1 June 2020)

• page 365, line 11: “|e^iθ -1| ≤ θ” should be “|e^iθ -1| ≤ |θ|”. (corrected 7 May 2021)

• page 377, second to last paragraph of proof of 11.87: “Part (a)” should be “Part (b)”. (corrected 1 June 2020)

• page 377, four times in last paragraph of proof of 11.87: “sp(T)” should be “sp(F )”. (corrected 28 August 2021)

• page 378, Exercise 9: “f ⁽ⁿ⁾ ∈ L¹(R)” should be “f ^(k) ∈ L¹(R) for k = 1, . . . , n”. (corrected 31 December 2020)

• page 379, Exercise 21: “that” should be “that if”. (corrected 20 November 2021)

• page 388, line 4: “|f_n(t)| ≤ t ” should be “|f_n(t)| ≤ |t| ”. (corrected 30 July 2020)

• page 392, line 6 from bottom: “probability distribution” should be “distribution function”. (corrected 17 April 2022)

• page 398, Exercise 2: The word “sets” should be deleted. (corrected 13 December 2021)

• page 399, Exercise 14: “s ∈ R” should be “s ∈ [0,1]”. (corrected 29 December 2021)

If you notice additional errors in the book or have suggestions for improvements, please let me know by sending an email message to measure@axler.net.

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