Last updated: Aug 13, 2020

An introduction to measure theory

Terence Tao 2011; 206 pp; hardcover ISBN-10: 0-8218-6919-1

ISBN-13: 978-0-8218-6919-2

Graduate Studies in Mathematics, vol. 126

This continues my series of books derived from my blog. The preceding books in this series were “Structure and Randomness“, “Poincaré’s legacies“, and “An epsilon of room“. It is based primarily on these lecture notes.

An online version of the text can be found here. The official AMS page for the book is here. There is no solution guide for this text.

The book has been reviewed for the American Mathematical Monthly by Takis Konstantopoulos, and also reviewed for the Mathematical Association of America by Mihaela Poplicher.

Errata:

Page 2: In the first paragraph, “same area” should be “same measure”.

Page 6: In Remark 1.1.3, “Exercise!” should be “exercise!”.

Page 8: In the first paragraph, “or even a rotated box” should be a separate sentence: “A rectangle with sides parallel to the axes is elementary, but most rotations of that rectangle will not be.”

Page 10: In Exercise 1.1.14, “epsilon entropy” is a slightly more accurate description here than “metric entropy”.

Page 11: In Exercise 1.1.19, add “Generalise this result to the case when is Jordan measurable instead of elementary”.

is Jordan measurable instead of elementary”. Page 14: In Exercise 1.1.24(3), “Jordan measurable of” should be “Jordan measurable subset of”.

Page 15: In Section 1.2, (iii), “inner and Jordan outer” should be “Jordan inner and outer”.

Page 17: Exercise 1.1.13 should be Exercise 1.1.5. In the last paragraph, “In the notes below” should be “In the rest of this section”.

Page 20: In the first paragraph, “Vol I” should be “Vol. I”

Page 21 Remark 1.2.7: “proof this” should be “proof of this”.

Page 27: In the proof of Lemma 1.2.13(v), (iv) should be (vi). In the proof of Lemma 1.2.13(vi), the phrases “By countable additivity” and “this implies that is contained ” should be interchanged.

is contained ” should be interchanged. Page 29: In the hint for Exercise 1.2.10: “conclude that is homeomorphic …” should be “conclude that the set of endpoints of the intervals is homeomorphic …”.

is homeomorphic …” should be “conclude that the set of endpoints of the intervals is homeomorphic …”. Page 32: In Exercise 1.2.13(ii), insert “Let , be as in part (i).”

, be as in part (i).” Page 34: In Exercise 1.2.22(i), “Lebesgue measure” should be “Lebesgue outer measure”.

Page 35: In Exercise 1.2.24(i), “a equivalence” should be “an equivalence”.

Page 36: In Exercise 1.2.25 and the following paragraph, “continuously differentiable” may be weakened to “Lipschitz continuous”.

Page 40: Near the end of the second paragraph, the reference to should be deleted.

should be deleted. Page 42: On line 12, “indicator function of these sets” should be “indicator functions of these sets”. In Definition 1.3.3, “a unsigned” should be “an unsigned”.

Page 45: In Definition 1.3.6, “said to be absolutely integrable of” should be “said to be absolutely integrable if”. Before this definition, “absolutely Lebesgue” should be “absolutely convergent Lebesgue”.

Page 46: In the hint for Exercise 1.3.2, “the second inequality” should be “the second equality”.

Page 52: In Exercise 1.3.8, both (iii) and (iv): “an” should be “a”. After Exercise 1.3.8, add the following question: Suppose that is measurable, and is a surjective linear map. Show that is also measurable. (Hint: uses Exercises 1.2.21 and 1.2.22.) What happens if the requirement that be surjective is dropped?”.

is measurable, and is a surjective linear map. Show that is also measurable. (Hint: uses Exercises 1.2.21 and 1.2.22.) What happens if the requirement that be surjective is dropped?”. Page 58: In Exercise 1.3.21, “greatest integer less than” should be “greatest integer less than or equal to”.

Page 60: At the end of Theorem 1.3.20, add “We call a function compactly supported if its support is contained in a compact set.”

Page 64: In the proof of Theorem 1.3.28, and should both be , should be , and the sentence fragment “, and the same is true for local uniform limits (because continuity is a local property)” should be deleted.

and should both be , should be , and the sentence fragment “, and the same is true for local uniform limits (because continuity is a local property)” should be deleted. Page 67: In Definition 1.4.1, “ of ” should be “ of subsets of “. “a sub-algebra of” should be moved from the fragment “ is finer than…” to “ is coarser than”.

of ” should be “ of subsets of “. “a sub-algebra of” should be moved from the fragment “ is finer than…” to “ is coarser than”. Page 68, Example 1.4.7: “finer… atomic algebra” should be “finer … atomic algebras”. At the end of Example 1.4.4, the period should be outside the parenthesis.

Page 70, Exercise 1.4.9, (ii): “either” should be “are either”.

Page 72, line 1: “only holds if and only if” should be “holds if and only if”.

Page 73, Remark 1.4.17: “so that ” should be “so that is the Borel -algebra”. In Exercise 1.4.15, should be .

” should be “so that is the Borel -algebra”. In Exercise 1.4.15, should be . Page 74, Section 1.4.3, l. 2: “a sigma-algebra a measurable space” should be “a measurable space”. In Remark 1.4.18, delete the left parenthesis before “Indeed”.

Page 75: In Exercise 1.4.20, “Boolean -algebra” should be “Boolean algebra”.

-algebra” should be “Boolean algebra”. Page 77: In Example 1.4.29, “Exercise 1.4.22” should be “Example 1.4.22”.

Page 81: In Definition 1.4.31 and Exercise 1.4.32, “a measurable space ” should be “a measure space “. In Exercise 1.4.33 (iv), the reference to Exercise 1.3.2 instead of Exercise 1.1.2.

” should be “a measure space “. In Exercise 1.4.33 (iv), the reference to Exercise 1.3.2 instead of Exercise 1.1.2. Page 83: In Exercise 1.4.35 (ix,x), “Horizontal” and “Vertical” should be interchanged.

Page 84: In the proof of Theorem 1.4.37, “horizontal” and “vertical” should be interchanged.

Page 85: In Exercise 1.4.39, “Exercise 1.4.26” should be “Example 1.4.26”.

Page 86: After Definition 1.4.38, add to the following paragraph “Clearly, this definition…” the sentence “As in that definition, one can extend the integral to measurable functions that are -almost everywhere defined, rather than everywhere defined.”

-almost everywhere defined, rather than everywhere defined.” Page 87: Replace the second half of the last sentence of Example 1.4.40 by “but the support of the are becoming increasingly wide, and so Exercise 1.4.41 does not apply”. In Example 1.4.41, “converges pointwise to ” should be “converges pointwise to “.

are becoming increasingly wide, and so Exercise 1.4.41 does not apply”. In Example 1.4.41, “converges pointwise to ” should be “converges pointwise to “. Page 88: In the proof of Theorem 1.4.43, “vertical truncation” should be “horizontal truncation”.

Page 91: In the first paragraph of the proof of Theorem 1.4.48, should be .

should be . Page 97: The final sentence of Remark 1.5.6 is redundant (it already appears in page 96) and can be deleted.

Page 99: In the fourth line of Section 1.5.2, “a measurable set” should be “an indicator function of a measurable set”.

Page 100: In Exercise 1.5.3(iii), replace the condition after “if and only if” by “ as “. Similarly in (vi), replace the condition after “if and only if” by “ as “.

as “. Similarly in (vi), replace the condition after “if and only if” by “ as “. Page 103, Section 1.5.5, line 4: “examples shows” should be “examples show”. In the second paragraph of Section 1.5.5, should take values in , rather than .

should take values in , rather than . Page 106: In the second display after (1.17), should just be .

should just be . Page 107: In Exercise 1.5.19, a comma is missing between “almost uniformly” and “pointwise”.

Page 108, line 5: a right parenthesis is missing before “is commonly used”. At the start of Section 1.6, add “Throughout this section, the notions of measurability and “almost everywhere” are understood to be with respect to Lebesgue measure.”

Page 112: For Theorem 1.6.11 and Exercise 1.6.5, “definite integral” should be “indefinite integral” (because the endpoint is allowed to vary).

is allowed to vary). Page 114 3rd paragraph, line 3: the symbol should be an . In the third display from bottom, should be . In the proof of Proposition 1.6.13, “Applying Littlewood’s second principle … to … ” should be “Applying Littlewood’s second principle … to … “.

should be an . In the third display from bottom, should be . In the proof of Proposition 1.6.13, “Applying Littlewood’s second principle … to … ” should be “Applying Littlewood’s second principle … to … “. Pages 115-116, Exercise 1.6.9: The second item here should be labeled (ii) (and the third should be labeled (iii)). In Remark 1.6.15, “equal to 2” should be “equal to 4”.

Page 117: In the paragraph after (1.24), “ is sufficiently close to ” should be “ is sufficiently close to “.

is sufficiently close to ” should be “ is sufficiently close to “. Page 118: In Lemma 1.6.17(ii), should be , and similarly for the second display after (1.25).

should be , and similarly for the second display after (1.25). Page 119: Near the bottom of the page, “Corollary 1.6.5” should be “Exercise 1.6.5”. In the second paragraph, replace “but not ” with “but is disjoint from (since for all )”, and should be .

” with “but is disjoint from (since for all )”, and should be . Page 120, Exercise 1.6.13: “Lemma 1.6.16” should be “Exercise 1.6.12”, and the hypothesis should be added.

should be added. Page 121: Before Exercise 1.6.14, “Lebesgue point for ” should be “Lebesgue point for “.

” should be “Lebesgue point for “. Page 122: In Remark 1.6.21, the fragment should be deleted. In Theorem 1.6.20, the second integral should be over rather than .

should be deleted. In Theorem 1.6.20, the second integral should be over rather than . Page 124: In the first and second displays, the integrals over should instead be over .

should instead be over . Page 125, Exercise 1.6.21: “Besicovich” should be “Besicovitch”; part (i) should be and as opposed to and . In part (ii) of this exercise, should be . In the hint for the exercise, “the the” should be “the”. In Exercise 1.6.22, “positive length” should be “positive finite length”. In Exercise 1.6.20, the integral over should instead be over .

and as opposed to and . In part (ii) of this exercise, should be . In the hint for the exercise, “the the” should be “the”. In Exercise 1.6.22, “positive length” should be “positive finite length”. In Exercise 1.6.20, the integral over should instead be over . Page 127: In Exercise 1.6.27(iii), add the parenthetical “In fact one can take .”

.” Page 128, Section 1.6.3, line 4: “continuous not differentiable” should be “continuous but not differentiable”. In Exercise 1.6.28(ii), delete “8-dyadic”, and replace “n” with “m” throughout to reduce confusion. Also, replace with , and replace with throughout the exercise.

with , and replace with throughout the exercise. Page 129: In part (iv), “lower right derivative” should be “lower left derivative”. Afterwards, “rather than on the endpoints” should be “rather than on the real line”. After Exercise 1.6.30, “four derivatives” should be “four Dini derivatives”.

Page 130: In the second to last line (in the proof of Lemma 1.6.26), should be . Similarly, on page 132 in the proof of Lemma 1.6.28, should be .

should be . Similarly, on page 132 in the proof of Lemma 1.6.28, should be . Page 131-132: and should be and respectively throughout. In the second display, should be . At the end of Exercise 1.6.31, add a right parenthesis. In the paragraph preceding Definition 1.6.30, remove the period after Lemma 1.6.26.

and should be and respectively throughout. In the second display, should be . At the end of Exercise 1.6.31, add a right parenthesis. In the paragraph preceding Definition 1.6.30, remove the period after Lemma 1.6.26. Page 133: In the proof of Lemma 1.6.31, should be .

should be . Page 134: On the eighth line: “ is discontinuous” should be “ is discontinuous”.

is discontinuous” should be “ is discontinuous”. Page 135: Before Definition 1.6.33, “absolutely convergent functions” should be “absolutely integrable functions”. After the first display, “four Dini derivatives” should be “three Dini derivatives”. In Definition 1.6.33, should be , and similarly on p. 136, 137.

should be , and similarly on p. 136, 137. Page 137: In the second paragraph, “it suffices to (by writing to show that ” should be “it suffices (by writing ) to show that “

to show that ” should be “it suffices (by writing ) to show that “ Page 141: in the definition (i) after Remark 1.6.38, “contains ” should be “whose closure contains “.

” should be “whose closure contains “. Page 144: In the third paragraph of the proof of the rising sun lemma (Lemma 1.6.17), should be in the definition of and in the next two occurrences (i.e. “ but not ” should be but not “, and “ ” should be “.

should be in the definition of and in the next two occurrences (i.e. “ but not ” should be but not “, and “ ” should be “. Page 145, bottom: “ exists” should be “ exists”. After Exercise 1.6.52, “ensure the almost everywhere existence” should be “ensure the absolute integrability of the derivative”.

exists” should be “ exists”. After Exercise 1.6.52, “ensure the almost everywhere existence” should be “ensure the absolute integrability of the derivative”. Pages 149-152: In Section 1.7.1, “Caratheodory extension theorem” should be “Caratheodory lemma” throughout.

Page 150, Exercise 1.7.2: “Lebesgue outer measurable” should be ” the Lebesgue outer measure”

Page 151: In the last two displays, and in the first display on the next page, may be simplified to . In the second paragraph, “a disjoint sequence of” should be “a sequence of disjoint”.

may be simplified to . In the second paragraph, “a disjoint sequence of” should be “a sequence of disjoint”. Page 156: In Theorem 1.7.9, should be . In the second paragraph of proof of this theorem, before “, adopting the obvious conventions”, add “to be the required value of given by (1.33) (e.g., )”.

should be . In the second paragraph of proof of this theorem, before “, adopting the obvious conventions”, add “to be the required value of given by (1.33) (e.g., )”. Page 157: Before (1.35), replace “By subadditivity, it suffices to show that” with “By finite additivity, we have for any , so it suffices to show that”. In the second display after (1.35), the right-hand side should be rather than . In the second and third paragraphs, “Exercise!” should be “exercise!”.

for any , so it suffices to show that”. In the second display after (1.35), the right-hand side should be rather than . In the second and third paragraphs, “Exercise!” should be “exercise!”. Page 158: Throughout this page, “Exercise!” should be “exercise!”.

Page 159: In Exercise 1.7.13, add a right parenthesis after “absolutely integrable”.

Page 160: In Exercise 1.7.14(ii), “delta functions” should be “Dirac measures” for consistency. In the first line, should be .

should be . Page 161: In Exercise 1.7.18 (i), $latex Y \in B_Y$ should be . In the second display, should be

. In the second display, should be Page 162: Exercise 1.7.19(ii) is not correct as stated and should be deleted.

as stated and should be deleted. Page 163: In the third paragraph of the proof of Proposition 1.7.11, should be .

should be . Page 165, Exercise 1.7.21: Add the line: “In particular, and are isomorphic as measure spaces and can thus safely be denoted as .” In the definition of a monotone class, “is a collection” should be “to be a collection”. In Example 1.7.13, should be . In Exercise 1.7.21, “ -finite sets” should be “ -finite measure spaces”.

and are isomorphic as measure spaces and can thus safely be denoted as .” In the definition of a monotone class, “is a collection” should be “to be a collection”. In Example 1.7.13, should be . In Exercise 1.7.21, “ -finite sets” should be “ -finite measure spaces”. Page 167: The sentence preceding Theorem 1.7.18 should be deleted.

Page 168, in (1.37), the third integral should have X and Y interchanged (as well as the measures and ).

and ). Page 169: In Exercise 1.7.22, “the counting measure (…) ” should be “the counting measure (…)”. In the second line of (1.38), the integral should be over rather than , and should be . In the seventh line from the bottom, “equal to one for every ” should be “equal to one for every “.

” should be “the counting measure (…)”. In the second line of (1.38), the integral should be over rather than , and should be . In the seventh line from the bottom, “equal to one for every ” should be “equal to one for every “. Page 170: In Exercise 1.7.23, the right-hand side of the display should read rather than . Also, “exist and are absolutely integrable” should be “exist as absolutely integrable integrals” (two occurrences). In the statement of Theorem 1.7.21(iii), the second appearance of should instead be . In Remark 1.7.22, “ -finite setting” should be “non- -finite setting”.

rather than . Also, “exist and are absolutely integrable” should be “exist as absolutely integrable integrals” (two occurrences). In the statement of Theorem 1.7.21(iii), the second appearance of should instead be . In Remark 1.7.22, “ -finite setting” should be “non- -finite setting”. Page 171: In Exercise 1.7.24, “Show that if” should be “Show that”.

Page 175: In the last complete paragraph, “for thus purpose” should be “for this purpose”.

Page 187: In the display before Remark 2.2.3, should be .

should be . Page 188: After (2.2), should just be .

should just be . Page 189: In the second paragraph, a comma is missing between “For ” and “ is clearly”. In the third paragraph, “ is a null set” should be “ is a null set”. In the fourth paragraph, “ is rational” should be “ is countable”.

” and “ is clearly”. In the third paragraph, “ is a null set” should be “ is a null set”. In the fourth paragraph, “ is rational” should be “ is countable”. Page 194: In the final sentence of Section 2.3, should be for notational consistency.

should be for notational consistency. Page 195: In Exercise 2.4.1(3), should be . In Exercise 2.4.1(8), should be .

should be . In Exercise 2.4.1(8), should be . Page 197: In the final display, should be defined as rather than . On the first display of the next page, the first occurrence of should be , and the final should just be . In the seventh paragraph, should be .

should be defined as rather than . On the first display of the next page, the first occurrence of should be , and the final should just be . In the seventh paragraph, should be . Page 205: The index entry for “restriction (measure)” should point to Example 1.4.25 rather than Exercise 1.4.35 (which could instead be referenced by “restriction (function)”.

Thanks to Daniel Barter, Andres Caicedo, alan Chang, Daniel Dorani, Robin Fissum, Stephen Ge, Travis Gibson, Yaver Gulusoy, Petri Henrik, Joe Li, Weifeng Li, Yoshiki Otobe, Arpan Pal, Navan Ranasinge, Xiao Shen, Daniel Shved, Frieder Simon, Isaac Solomon, Gandhi Viswanathan, Deven Ware, Ittay Weiss, and Luqing Ye for corrections.