Last updated: Oct 2, 2014

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 11: In Exercise 1.1.19, add “Generalise this result to the case when $F$ 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 $\bigcup_{n=1}^\infty E_n$ is contained $\bigcup_{n=1}^\infty U_n$” should be interchanged.
• 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 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 67: In Definition 1.4.1, “$B$ of $X$” should be “$B$ of subsets of $X$“.
• 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 $\langle {\mathcal F} \rangle$” should be “so that $\langle {\mathcal F}$ is the Borel $\sigma$-algebra”.  In Exercise 1.4.15, ${\mathcal F}_{n-1}$ should be ${\mathcal F}_\alpha$.
• Page 74, Section 1.4.3, l. 2: “a sigma-algebra a measurable space” should be “a measurable space”.
• Page 83: In Exercise 1.4.35 (ix,x), “Horizontal” and “Vertical” should be interchanged.
• Page 97: The final sentence of Remark 1.5.6 is redundant (it already appears in page 96) and can be deleted.
• Page 103, Section 1.5.5, line 4: “examples shows” should be “examples show”.
• Page 106: In the second display after (1.17), $\leq \varepsilon \leq$ should just be $\leq$.
• Page 108, line 5: a right parenthesis is missing before “is commonly used”.
• Page 114 3rd paragraph, line 3: the symbol $F'$ should be an $f$.  In the third display from bottom, $(f_h-f)_h$ should be $(f_h-f)$.
• 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 120, Exercise 1.6.13: “Lemma 1.6.16″ should be “Exercise 1.6.12″, and the hypothesis $\lambda>0$ should be added.
• Page 125, Exercise 1.6.21: “Besicovich” should be “Besicovitch”; part (i) should be $I_i$ and $I_j$ as opposed to $I_n$ and $I_m$.
• Page 128, Section 1.6.3, line 4: “continuous not differentiable” should be “continuous but not differentiable”.
• Page ???: In the proof of Proposition 1.6.13, “Applying Littlewood’s second principle … to … $F'$” should be “Applying Littlewood’s second principle … to … $f$“.
• Page ???: In the paragraph after (1.24), “$h$ is sufficiently close to $x$” should be “$h$ is sufficiently close to $0$“.
• Page 144: In the third paragraph of the proof of the rising sun lemma  (Lemma 1.6.17), $b$ should be $b_n$ in the definition of $A$ and in the next two occurrences (i.e. “$t$ but not $b$” should be $t$ but not $b_n$“, and “$t_* \in [t,b)$” should be $t_* \in [t,b_n)$“.
• Page 145, bottom: “$f'(x)$ exists” should be “$F'(x)$ exists”.
• Page 150, Exercise 1.7.2: “Lebesgue outer measurable” should be ” the Lebesgue outer measure”
• Pages ???: In Section 1.7.1, “Caratheodory extension theorem” should be “Caratheodory lemma” throughout.
• Page 157: Before (1.35), replace “By subadditivity, it suffices to show that” with “By finite additivity, we have $\mu_0(E) \geq \sum_{n=1}^N \mu_0(E_n)$ for any $N$, so it suffices to show that”.  In the second display after (1.35), the right-hand side should be $\inf_{U \supset E_n} \mu_0(U)$ rather than $\inf_{U \supset E_n} \mu_0(E_n)$.
• Page 165, Exercise 1.7.21: Add the line: “In particular, $X \times (Y \times Z)$ and $(X \times Y) \times Z$ are isomorphic as measure spaces and can thus safely be denoted as $X \times Y \times Z$.”   In the definition of a monotone class, “is a collection” should be “to be a collection”.
• Page 168, in (1.37), the third integral should have X and Y interchanged (as well as the measures $d\mu_Y(y)$ and $d\mu_X(x)$).
• Page 169: In Exercise 1.7.22, “the counting measure (…) $\#$” should be “the counting measure $\#$ (…)”.
• Page 170: In Exercise 1.7.23, the right-hand side of the display should read $\int_{[0,1]} (\int_{[0,1]} f(x,y)\ dx)\ dy$ rather than $\int_{[0,1]} (\int_{[0,1]} f(x,y)\ dy)\ dx$.  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 $\int_X (\int_Y f(x,y)\ d\mu_Y(y))\ d\mu_X(x)$ should instead be  $\int_Y (\int_X f(x,y)\ d\mu_X(x))\ d\mu_Y(y)$.
• Page 197: In the final display, $K'_N$ should be defined as $\bigcap_{N'=1}^N \pi_{B_{N'} \leftarrow B_N}^{-1}(K_{N'})$ rather than $\bigcup_{N'=1}^N \pi_{B_{N'} \leftarrow B_N}^{-1}(K_{N})$.  On the first display of the next page, the first occurrence of $\varepsilon/2^N$ should be $\sum_{N'=1}^N \varepsilon/2^{N'}$, and the final $\varepsilon - \varepsilon/2^N$ should just be $\varepsilon/2^N$.
• Page ???: In the display before Remark 2.2.3, $h \to {\bf R}^d \backslash \{0\}$ should be $h \in {\bf R}^d \backslash \{0\}$.
Thanks to Daniel Barter, Andres Caicedo, Stephen Ge, Travis Gibson, Petri Henrik, Xiao Shen, Daniel Shved, Isaac Solomon, Gandhi Viswanathan, Deven Ware, Ittay Weiss, and Luqing Ye for corrections.