Last updated: Sep 27, 2015
An introduction to measure theory
Terence Tao2011; 206 pp; hardcover
ISBN10: 0821869191
ISBN13: 9780821869192
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 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”.
 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 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.
 Page 32: In Exercise 1.2.13(ii), insert “Let , 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 40: Near the end of the second paragraph, the reference to 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”.
 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 67: In Definition 1.4.1, “ of ” should be “ of subsets of “. “a subalgebra 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”.
 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 .
 Page 74, Section 1.4.3, l. 2: “a sigmaalgebra a measurable space” should be “a measurable space”.
 Page 77: In Example 1.4.29, “Exercise 1.4.22” should be “Example 1.4.22”.
 Page 83: In Exercise 1.4.35 (ix,x), “Horizontal” and “Vertical” should be interchanged.
 Page 85: In Exercise 1.4.39, “Exercise 1.4.26” should be “Example 1.4.26”.
 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”.
 Page 91: In the first paragraph of the proof of Theorem 1.4.48, 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 “.
 Page 103, Section 1.5.5, line 4: “examples shows” should be “examples show”.
 Page 106: In the second display after (1.17), 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).
 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 … “.
 Pages 115116, 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 “.
 Page 120, Exercise 1.6.13: “Lemma 1.6.16” should be “Exercise 1.6.12”, and the hypothesis should be added.
 Page 122: In Remark 1.6.21, the fragment should be deleted.
 Page 125, Exercise 1.6.21: “Besicovich” should be “Besicovitch”; part (i) should be and as opposed to and .
 Page 128, Section 1.6.3, line 4: “continuous not differentiable” should be “continuous but not differentiable”. In Exercise 1.6.28(ii), delete “8dyadic”, and replace “n” with “m” throughout to reduce confusion.
 Page 131132: and should be and respectively throughout.
 Page 133: In the proof of Lemma 1.6.31, should be .
 Page 134: On the eighth line: “ is discontinuous” should be “ is discontinuous”.
 Page 137: In the second paragraph, “it suffices to (by writing to show that ” should be “it suffices (by writing ) to show that “
 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 “.
 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”.
 Pages 149152: 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 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 righthand side should be rather than .
 Page 160: In Exercise 1.7.14(ii), “delta functions” should be “Dirac measures” for consistency.
 Page 161: In Exercise 1.7.18 (i), $latex Y \in B_Y$ should be .
 Page 162: Exercise 1.7.19(ii) is not correct as stated and should be deleted.
 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”.
 Page 166: In the second paragraph, “ contains ” should be “ contains “.
 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 ).
 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 “.
 Page 170: In Exercise 1.7.23, the righthand 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 “nonfinite setting”.
 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 .
 Page 188: After (2.2), 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”.
 Page 194: In the final sentence of Section 2.3, should be for notational consistency.
 Page 195: In Exercise 2.4.1(3), 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 .
 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)”.
45 comments
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24 January, 2011 at 8:35 am
An introduction to measure theory « What’s new
[…] theory notes to book form, a draft copy of which is now available here. I have also started up a stub of a book page for this text, though it has little content at present beyond that link. I will be continuing […]
12 March, 2013 at 3:13 am
Juan Kuntz
Dear Professor Tao,
Thank you very much for posting the book online – it’s great. I’ve been working independently through the exercises and I can do an ok amount on my own fine – however, there are some in which I feel I would really benefit from the input of others. As I a rarely come in contact with people who would be interested in such exercises (I’m not a mathematician by training, nor I’m based in a mathematics department), I was considering attempting to start some sort of (obviously free) collaborative online solution manual where readers could post and discuss solutions to the exercises, e.g., something along the lines of http://www.roadtoreality.info/.
Would you approve of such a thing? Of course, if you do not, I will drop it.
12 March, 2013 at 12:30 pm
Terence Tao
I have no objection to this, though it may be that the number of participants in such a project could be rather low.
13 March, 2013 at 7:29 am
Juan Kuntz
Possibly, probably still worth a shot though. Thank you.
12 November, 2013 at 3:19 pm
Richard Schur
Dear Juan,
How is the website going so far?
Sincerely,
Richard Schur
19 January, 2015 at 3:01 pm
cjpn
Dear Professor Tao,
Like Juan, I am not a mathematician by training, but I am interested in learning about measure theory and have just started working through your book. If anyone else is interested I am posting my answers to the Exercises on my blog (https://cjpnmiscellany.wordpress.com/) and would welcome any comments or suggestions for improved solutions.
25 January, 2011 at 12:20 am
Dirk
Are you sure about the spelling “Radamacher” vs “Rademacher”? I am not totally sure either, but I think it is Hans Rademacher (however, I could not find his particular publication about the differentiation theorem).
[Oops – this will be corrected in the next revision. – T.]
26 January, 2011 at 9:11 am
Jason T. Miller
For my own purpose of reading on the iPad, I cropped and bookmarked the draft, and thought the result might be helpful to other readers:
http://mypage.iu.edu/~jasomill/misc/terrytao_measure_theory_draft_1101.pdf
Dr. Tao: if you don’t approve, please feel free to kill the comment — this is the only place I’ve linked to it (or intend to). And thanks for writing and posting this material in the first place!
14 February, 2011 at 1:23 pm
Anonymous
Hi Professor Tao,
You mention in the book (Remark 1.3.1) that one can first define the Lebesgue integral for bounded measurable nonnegative functions on sets of finite measure and then extend to the general case of nonnegative measurable functions (by taking the sup over the previously defined functions). Given that this requires two steps, and thus appears to be more complicated that just defining the integral as you have done in Definition 1.3.12, why would one choose to do this? You suggest in Remark 1.4.50 that it may come down to taste. Do you mean by this that some of the proofs become simpler if one introduces the added complexity in the definitions at the beginning?
Thanks.
14 February, 2011 at 1:58 pm
Terence Tao
Yes, there are some simplifications if one initially restricts the integral to bounded functions of finite measure support. Specifically, establishing additivity of the integral is easier because one does not need the horizontal and vertical truncation tools for such integrals. The two approaches are basically permutations of each other and have comparable net complexity.
3 March, 2011 at 3:56 pm
Sarah
What is the license on this manuscript? For example, if one were to want to use it for a class (at one’s own risk in terms of errata, of course), would that be permissible?
4 March, 2011 at 6:37 am
Terence Tao
The book (like my other blog books) is published conventionally by the AMS (with the copyright held by myself); they have agreed to permit a copy of the book to be hosted on this site, but this would be intended primarily for personal use (or as an alternative source for the physical text, if the latter is somehow unavailable). For use as a formal classroom text, I would imagine that the book form would be preferable.
6 March, 2011 at 9:00 am
Sarah
I was asking primarily since this is a draft manuscript and the printed book is not available yet. Of course once the book becomes available that will be preferable. When is the book scheduled to appear in print?
6 March, 2011 at 9:09 am
Terence Tao
Ah, I had confused this book (which has not yet been published) with my earlier blog books. But judging from previous experience, I would imagine that it would take about twelve months or so before the book is available in print.
10 March, 2011 at 11:18 am
Anonymous
Thanks! I am considering adopting the book for a course in the fall, or using it as one of two main resources, but I certainly would not want to do that without permission.
3 March, 2011 at 10:37 pm
Daniel
Dear Prof. Tao, your section on problemsolving strategies was a delight to read! Thank you for making it available.
May I ask where the concept of “giving yourself an epsilon of room” originated? Is it your own?
6 March, 2011 at 9:10 am
Terence Tao
The precise term is my own, but the idea of “epsilon regularisation” or the “density argument” has been around since the days of Lebesgue at least; I do not know who to first ascribe it to.
22 July, 2011 at 10:01 pm
Literaturempfehlung « UGroh's Weblog
[…] In diesem Buch hat T. Tao seine Blogmitteilungen zu dieser mathematischen Disziplin zusammengefasst. Ich persönlich finde, dass dieses eines der besten Bücher zur Maßtheorie ist, da die Theorie motivierend aufgebaut ist. Ein PDF des Buches findet sich auf seiner Buchseite (siehe seinen Blog). […]
15 September, 2011 at 10:39 am
“Introduction to measure theory” now published « What’s new
[…] graduate text on measure theory (based on these lecture notes) is now published by the AMS as part of the Graduate Studies in […]
16 September, 2011 at 9:02 am
Literaturempfehlung « UGroh's Weblog
[…] T. Tao nunmehr bestellbar ist. Da es auch als PDFFile verfügbar ist, kann sich jeder vorab auf dem Blog von T. Tao […]
13 February, 2012 at 11:40 pm
Daniel Shved
Dear Prof. Tao,
There is a phrase in the book that confused me a little. On page 32, in the proof for Lemma 1.2.13(vi), it says: “By countable subadditivity, this implies that is contained in , and the difference has Legesgue outer measure at most “. I see how the second part follows from countable subadditivity, but the fact that is contained in is true by itself, without countable subadditivity. Maybe it should be like this: “ is contained in , and, by countable subadditivity, the difference has Legesgue outer measure at most “? Or am I not getting something?
14 February, 2012 at 3:39 am
Daniel Shved
Oops, there are at least two typos in my comment above: Legesgue>Lebesgue, and > . Also, there’s something wrong with the way epsilons look… :(
16 February, 2012 at 10:22 pm
andrescaicedo
(Small typo.) In page 17, I think that the reference to Exercise 1.1.13 should instead be to Exercise 1.1.5.
[Correction added, thanks. T.]
17 February, 2012 at 4:46 pm
515 – Advanced Analysis « Teaching blog
[…] Graduate studies in mathematics, vol 126, 2011. ISBN10: 0821869191. ISBN13: 9780821869192. Errata. Mathematicians find it easier to understand and enjoy ideas which are clever rather than subtle. […]
25 February, 2012 at 12:13 pm
Daniel Shved
Dear prof. Tao,
here are several misprints:
p41, ex. 1.2.22 (i): Lebesgue measure, etc > Lebesgue outer measure, etc
p42, ex. 1.2.24 (i): Show this is a equivalence relation > Show this is an equivalence relation
p50, before section 1.3.1: The facts listed here manifestations > The facts listed here are manifestations
p54, before definition 1.3.6: absoutely Lebesgue integral > absolutely convergent Lebesgue integral
p56, hint for ex. 1.3.2: rearrange the second inequality > rearrange the second equality
p70, ex. 1.3.21 (recheck this): where [x] is the greatest integer less than x > less than or equal to x
p75, before the proof of Egorov’s theorem: since one local uniform convergence > since one has local uniform convergence (not sure about this one)
p88, ex. 1.4.15(ii): countable number of sets in > countable number of sets in
p97, before definition 1.4.34: we defined first an simple integral > we defined first a simple integral
p101, ex. 1.4.36. (ix) is called “vertical truncation” and (x) is called “horizontal”. It should be the other way around. (Or, if this is correct, then “vertical” and “horizontal” should be swapped everywhere in an earlier section about the Lebesgue unsigned integral).
[Corrections added, thanks. T.]
27 August, 2012 at 7:37 am
515 – Advanced Analysis « A kind of library
[…] Graduate studies in mathematics, vol 126, 2011. ISBN10: 0821869191. ISBN13: 9780821869192. Errata. Mathematicians find it easier to understand and enjoy ideas which are clever rather than subtle. […]
29 November, 2012 at 2:33 pm
Isaac Solomon
On page 170, Exercise 1.7.23 (regarding the failure of Fubini’s theorem when certain conditions are relaxed), the problem asks to prove the inequality of two double integrals, but these integrals are actually the same, i.e. the order of dy and dx are not exchanged in the latter integral.
Also, the wording of the problem itself is a bit peculiar, in that it asks to show that the single integrals are absolutely integrable for a particular value of x or y, rather than the functions being absolutely integrable, and also that the double integrals exist and are absolutely integrable, rather than the functions inside the double integral being absolutely double integrable.
[Corrected, thanks – T.]
27 January, 2013 at 7:34 pm
Jack
In the errata, page 33 should be page 27 in the book if I don’t get the wrong book. (page 33 is the page number of the online version.)
[Corrected, thanks – T.]
19 November, 2013 at 8:32 am
hp
In the proof of the Rising Sun Lemma on
Page 144, the b in the set A should be changed to b_n instead. Similarily the b on the next row should be b_n. On the row under that row the first b should be changed to b_n.
29 November, 2013 at 7:20 am
David
Hello! I saw that you proved the additivity of the Lebesgue integral for nonnegative measurable functions (Theorem 1.4.38), but I couldn’t find in your book that you proved the additivity for general Lebesgue integrable functions, yet it appears that you used it in the proof of the dominated convergence theorem (Theorem 1.4.49). I finally came up with a proof, nevertheless I would like to know how you would prove it, or whether it is already in your book. My method was to prove the additivity first for simple functions, which then gives a short proof for nonnegative functions using the monotone convergence theorem for nonnegative functions, and then proving for general integrable functions by using simple sequences that converge uniformly to them on a bounded set such that their integrals are small on its complement. Do you have a shorter proof?
29 November, 2013 at 8:22 am
Terence Tao
This is Exercise 1.4.41(ii) (which generalises Exercise 1.3.19). Hint: given realvalued functions , find an identity connecting the positive and negative parts of , and rearrange this identity so that Theorem 1.4.38 may be applied.
29 November, 2013 at 9:31 am
David
I got it. Thanks! By the way, what’s your reason for not using the monotone convergence theorem for sets to prove the monotone convergence theorem for nonnegative functions first, so that we wouldn’t have to use truncation?
1 December, 2013 at 12:37 pm
Jordan
Hi Professor Tao, I’m an algebraist who’s been working through your book to become more well rounded in pure mathematics. I’m getting through most of the exercises pretty well but am having trouble with one in particular. Do you have any tips on how to attack exercise 1.7.21? In particular, the part about the associativity of the sigma algebras.
1 December, 2013 at 2:38 pm
Terence Tao
When trying to establish properties of abstract sigma algebras, Remark 1.4.15 is often useful. (One should also be proving double containment of the sigma algebras rather than equality, as per Strategy 2.1.1.)
1 December, 2013 at 2:52 pm
Jordan
I understand the concept of proving the double containment, but I’m not sure I see how to apply the remark in this case.
1 December, 2013 at 5:40 pm
Terence Tao
As I said, the key is to use Remark 1.4.15 to obtain the required containments. You might first warm up with other exercises that use this remark, e.g. Exercise 1.4.17 or Exercise 1.4.18.
7 December, 2013 at 4:06 pm
Ultraproducts as a Bridge Between Discrete and Continuous Analysis  What's new
[…] of (say) the unit cube, and that of Lebesgue measure of Borel subsets of that cube; see e.g. this text of mine for the basic theory here. For instance, it is not difficult to show that every Loeb measurable set […]
9 May, 2014 at 11:58 am
Jack
I’ve seen several exercises similar to Ex 1.1.3 (Uniqueness of elementary measure):
Ex 1.1.24 (Basic properties of the Riemann integral)
Ex 1.2.23 (Uniqueness of Lebesgue measure)
etc.
Can we use the those properties to “define” the corresponding mathematical concepts? Is there any tradeoff if we do such sort of thing in every definition in the book?
(This seems very similar to the philosophy of defining the determinant of square real matrices. It seems to me that what really matters is the properties pertaining to the concepts.)
9 May, 2014 at 12:25 pm
Terence Tao
Yes, one can certainly define these concepts axiomatically rather than constructively (for instance, defining Lebesgue measure to be the unique Haar measure that gives the unit cube a measure of 1), although then the difficulty then shifts to proving existence of the objects so defined. In the case of Lebesgue measure, for instance, one now needs the existence of Haar measure, which is a somewhat nontrivial fact to establish (particularly if one has not yet proven the Riesz representation theorem).
23 May, 2014 at 3:40 pm
Frederik
I’m having troubles showing the inequality on page 197 – 198, that
I can show that
but I think that I have “lost” to much in the process in order to obtain the inequality you present. Can you (or somebody else reading this) give me a push in the right direction?
[See erratum for this inequality on this page – T.]
23 May, 2014 at 5:21 pm
Frederik
Thanks for your quick reply. I’m actually already working on showing the revised inequality. As I have understood your corrections I should be able to show that
However I’m only able to show the inequality I mentioned above, using that
iteratively. I have tried a number of different approaches without any luck, so I hope that you can help me in the right direction. Thanks a lot!
23 May, 2014 at 7:29 pm
Terence Tao
Cover by for .
3 June, 2014 at 3:22 pm
Anonymous
Dear Professor Tao,
Here are some misprints I found that are not currently on the list:
p.125, ex 1.6.21: Besicovich > Besicovitch; part (i) should be I_i and I_j as opposed to I_n and I_m
p.145, bottom of p.145, f'(x) exists > F'(x) exists
p.150, ex 1.7.2: Lebesgue outer measurable > the Lebesgue outer measure
p.168, in (1.37) the third integral should have X and Y interchanged
[Added, thanks – T.]
2 October, 2014 at 9:03 am
Travis
Hi,
p. 227 just above Remark 2.2.3 under the limit, the second $\to$ should be $\in$
[Correction added, thanks – T.]
28 October, 2014 at 6:13 pm
Anon
Dr. Tao,
Hello. Do you know if AMS will print a second edition any time soon?
[There are no plans for this as yet, though if a sufficient number of further errata are unearthed, this may change. T.]