Graduate Studies in Mathematics 126. 206 pp., hardcover.
This continues my series of books derived from my blog. It is based primarily on my 245A real analysis lecture notes. There is no solution guide for this text.
In general: Most occurrences of \ldots can be replaced with \dots. Throughout, “Exercise!” should be “exercise!”.
p. xi“adjointed” should be “adjoined”.
p. 2In the first paragraph, “same area” should be “same measure”. In the third paragraph, “$\infty \cdot 0$ is indeterminate” should be “$\infty \cdot 0$ vanishes by our conventions”
p. ?The hint for Exercise 0.0.2 can be replaced with a hint that one may find Exercise 0.0.1 to be useful.
p. 5In Exercise 1.1.1, the first “and” can be deleted.
p. ?In Lemma 1.1.2(ii), clarify that $N$ is restricted to the natural numbers.
p. 8In 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.”
p. 10In Exercise 1.1.14, “epsilon entropy” is a slightly more accurate description here than “metric entropy”.
p. 11In Exercise 1.1.19, add “Generalise this result to the case when $F$ is Jordan measurable instead of elementary”. In Exercise 1.1.18(4), the second $Q^2$ should be in boldface.
p. 14In Exercise 1.1.24(3), “Jordan measurable of” should be “Jordan measurable subset of”.
p. 15In Section 1.2, (iii), “inner and Jordan outer” should be “Jordan inner and outer”.
p. ?Before Example 1.2.1, “we will briefly review below” should be “we reviewed in the preface”.
p. 16in order to deal with the peculiarities of zero-dimensional Euclidean spaces (in which all boxes are non-empty), in the definition of Lebesgue outer measure, one should use at most countably many boxes, rather than countably infinite many boxes.
p. 17Exercise 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”.
p. 19“assigment” should be “assignment”, and “of element of” should be “of elements of”.
p. 20In the first paragraph, “Vol I” should be “Vol. I”
p. 21Remark 1.2.7: “proof this” should be “proof of this”.
p. 22after (1.2), “for every $E \subset {\mathbf R}^d$” should be “for every bounded $E \subset {\mathbf R}^d$“. A little earlier, “Applying by” should just be “Applying”
p. 27In 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.
p. 29In the hint for Exercise 1.2.10: “conclude that $[0,1)$ is homeomorphic …” should be “conclude that the set of endpoints of the intervals is homeomorphic …”, and “infinitely many closed intervals” should be “countably many closed intervals”.
p. 32In Exercise 1.2.13(ii), insert “Let $E_n$, $E$ be as in part (i).”
p. 34In Exercise 1.2.22(i), “Lebesgue measure” should be “Lebesgue outer measure”.
p. 35In Exercise 1.2.24(i), “a equivalence” should be “an equivalence”. In Exercise 1.2.24(ii), $m$ should be $m^*$ in $m(E \Delta E') = m(F \Delta F')$.
p. 36In Exercise 1.2.25 and the following paragraph, “continuously differentiable” may be weakened to “Lipschitz continuous”.
p. 40Near the end of the second paragraph, the reference to $h$ should be deleted. In the definition of the lower Darboux integral, a “$dx$” is missing.
p. 42On 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”.
p. ?In the proof of Lemma 1.3.4, “each of… arise as unions” -> “each of… arises as the union”.
p. 45In 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”.
p. 46In the hint for Exercise 1.3.2, “the second inequality” should be “the second equality”. In Exercise 1.3.2(iii), the set $E$ needs to have finite measure.
p. 48“each of which are bounded” should be “each of which is bounded”.
p. 50In Exercise 1.3.3(iii), “unsigned measurable functions” should be “at most countable family of unsigned measurable functions”.
p. 52In Exercise 1.3.8, both (iii) and (iv): “an” should be “a”. After Exercise 1.3.8, add the following question: Suppose that $f: {\bf R}^d \to {\bf C}$ is measurable, and $T: {\bf R}^{d'} \to {\bf R}^{d}$ is a surjective linear map. Show that $f \circ T: {\bf R}^{d'} \to {\bf C}$ is also measurable. (Hint: uses Exercises 1.2.21 and 1.2.22.) What happens if the requirement that $T$ be surjective is dropped?”.
p. 52Exercise 1.3.9 should be moved to after Lemma 1.3.15 (Markov’s inequality), and the following hint added: “Show that $f$ can be expressed as the increasing pointwise limit of piecewise constant functions $g_n$, as well as a decreasing pointwise limit of piecewise constant functions $h_n$, with the Riemann integral of $h_n - g_n$ converging to zero. Using Markov’s inequality, show that the difference $h-g$ between the limit $h$ of the $h_n$ and the limit $g$ of $g_n$ is then zero almost everywhere.
p. 54In Exercise 1.3.10(x), “i.e.” should be “in particular, “.
p. 58In Exercise 1.3.21, “greatest integer less than” should be “greatest integer less than or equal to”.
p. 60At the end of Theorem 1.3.20, add “We call a function compactly supported if its support is contained in a compact set.”
p. 64In the proof of Theorem 1.3.28, $2^{n+1}$ and $2^{n-1}$ should both be $2^n$, $\varepsilon/2$ should be $\varepsilon$, and the sentence fragment “, and the same is true for local uniform limits (because continuity is a local property)” should be deleted.
p. 67In Definition 1.4.1, “$B$ of $X$” should be “$B$ of subsets of $X$“. “a sub-algebra of” should be moved from the fragment “${\mathcal B}'$ is finer than…” to “${\mathcal B}$ is coarser than”.
p. ?In Exercise 1.4.3, “if and only if exists a bijection” should be “if and only if there exists a bijection”. In Exercise 1.4.2, “length” should be sidelength”.
p. 68Example 1.4.7: “finer… atomic algebra” should be “finer … atomic algebras”, and “one or more” should be “zero or more”. At the end of Example 1.4.4, the period should be outside the parenthesis.
p. 70Exercise 1.4.9, (ii): “either” should be “are either”. In Exercise 1.4.7, ${\mathcal E}[{\mathbf R}^d]$ should be $\overline{{\mathcal E}[{\mathbf R}^d]}$.In Definition 1.4.12, ${\mathcal B}$ should be a collection of *subsets of* $X$.
p. 72line 1: “only holds if and only if” should be “holds if and only if”.
p. 73Remark 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 ${\mathcal F}_\alpha$.
p. 74Section 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”.
p. 75In Exercise 1.4.20, “Boolean $\sigma$-algebra” should be “Boolean algebra”.
p. ?In Exercise 1.4.24, “converge” should be “converges”.
p. 77In Example 1.4.29, “Exercise 1.4.22” should be “Example 1.4.22”.
p. 81In Definition 1.4.31 and Exercise 1.4.32, “a measurable space $(X, {\mathcal B})$” should be “a measure space $(X, {\mathcal B}, \mu)$“. In Exercise 1.4.33 (iv), the reference to Exercise 1.3.2 instead of Exercise 1.1.2. The definition of almost everywhere should be moved to before Exercise 1.4.31 where it is first used.
p. 83In Exercise 1.4.35 (ix,x), “Horizontal” and “Vertical” should be interchanged.
p. 84In the proof of Theorem 1.4.37, “horizontal” and “vertical” should be interchanged.
p. 85In Exercise 1.4.39, “Exercise 1.4.26” should be “Example 1.4.26”.
p. 86After 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 $\mu$-almost everywhere defined, rather than everywhere defined.”
p. 87Replace the second half of the last sentence of Example 1.4.40 by “but the support of the $f_n$ are becoming increasingly wide, and so Exercise 1.4.41 does not apply”. In Example 1.4.41, “converges pointwise to $f$” should be “converges pointwise to $f := 0$“.
p. 88In the proof of Theorem 1.4.43, “vertical truncation” should be “horizontal truncation”.
p. 91In the first paragraph of the proof of Theorem 1.4.48, $|f_n$ should be $|f_n|$.
p. ?In Exercise 1.4.51, add a remark that in this exercise the simple functions should be signed rather than unsigned.
p. 96In Exercise 1.5.1, $|g_n| \leq f_n$ may be replaced by $|g_n| \leq |f_n|$.
p. 97The final sentence of Remark 1.5.6 is redundant (it already appears in page 96) and can be deleted. Also, “essential upper bounds for $f$” should be “essential upper bounds for $|f|$“.
p. 99In the fourth line of Section 1.5.2, “a measurable set” should be “an indicator function of a measurable set”.
p. 100In Exercise 1.5.3(iii), replace the condition after “if and only if” by “$\min(A_n,\mu(E^*_n)) \to 0$ as $n \to \infty$“. Similarly in (vi), replace the condition after “if and only if” by “$\min(A_n,\mu(E_n)) \to 0$ as $n \to \infty$“. In 1.5.3 (vii) “converges in $L^1$ norm” should be “converges in $L^1$ norm to zero”. Before Remark 1.5.8, in part (iii), add the word “scenario” for consistency with (i) and (ii).
p. 103Section 1.5.5, line 4: “examples shows” should be “examples show”. In the second paragraph of Section 1.5.5, $g$ should take values in $[0,+\infty]$, rather than ${\bf C}$. In Exercise 1.5.9, “using Exercise 1.5.6” should be “using Exercise 1.5.8”.
p. 104In Exercise 1.5.10, the dominated convergence theorem should be used instead of the monotone convergence theorem.
p. ?The definition of uniform integrability in Definition 1.5.11 is a little weaker than the commonly accepted one in the case of infinite measure spaces. The blog post associated to this section contains the corrected version (with attendant changes to some subsequent material).
p. 106In the second display after (1.17), $\leq \varepsilon \leq$ should just be $\leq$.
p. 107In Exercise 1.5.19, a comma is missing between “almost uniformly” and “pointwise”.
p. 108line 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.”
p. 112For Theorem 1.6.11 and Exercise 1.6.5, “definite integral” should be “indefinite integral” (because the endpoint $x$ is allowed to vary). In Theorem 1.6.11, $[-\infty,x]$ should be $(-\infty,x]$.
p. 1143rd paragraph, line 3: the symbol $F'$ should be an $f$. In the third display from bottom, $(f_h-f)_h$ should be $(f_h-f)$. In the proof of Proposition 1.6.13, “Applying Littlewood’s second principle … to … $F'$” should be “Applying Littlewood’s second principle … to … $f$“.
p. 115-116Exercise 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”, and “if one force” should be “if one wishes to force”.
p. 117In the paragraph after (1.24), “$h$ is sufficiently close to $x$” should be “$h$ is sufficiently close to $0$“.
p. 118In Lemma 1.6.17(ii), $[a,b]$ should be $(a,b]$, and similarly for the second display after (1.25).
p. 119Near the bottom of the page, “Corollary 1.6.5” should be “Exercise 1.6.5”. In the second paragraph, replace “but not $b$” with “but is disjoint from $[b_n,b]$ (since $F(y) \leq F(b_n) < F(t)$ for all $b_n \leq y \leq b$)”, and $t_* \in [t,b)$ should be $t_* \in [t,b_n)$.
p. 120Exercise 1.6.13: “Lemma 1.6.16” should be “Exercise 1.6.12”, and the hypothesis $\lambda>0$ should be added.
p. 121Before Exercise 1.6.14, “Lebesgue point for ${\bf R}^d$” should be “Lebesgue point for $f$“.
p. ?In Exercise 1.6.20, Exercise 1.1.14 should be Lemma 1.2.11.
p. 122In Remark 1.6.21, the fragment $\geq \lambda \}$ should be deleted. In Theorem 1.6.20, the second integral should be over ${\bf R}^d$ rather than ${\bf R}$.
p. 124In the first and second displays, the integrals over ${\bf R}$ should instead be over ${\bf R}^d$.
p. 125Exercise 1.6.21: “Besicovich” should be “Besicovitch”; part (i) should be $I_i$ and $I_j$ as opposed to $I_n$ and $I_m$. In part (ii) of this exercise, $I'_m$ should be $I'_j$. 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 ${\bf R}$ should instead be over ${\bf R}^d$.
p. 127In Exercise 1.6.27(iii), add the parenthetical “In fact one can take $C'_d = 1$.”
p. ?In the proof of Lemma 1.6.28, “$E_{r,R}$ is contained” should be “$E_{r,R} \cap (a,b)$ is contained”
p. 128Section 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 $\sin(8^n \pi x)$ with $\cos(16^n \pi x)$, and replace $8$ with $16$ throughout the exercise.
p. 129In 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”.
p. 130In the second to last line (in the proof of Lemma 1.6.26), $G(b_n) \leq G(a_n)$ should be $G(b_n) \geq G(a_n)$. Similarly, on page 132 in the proof of Lemma 1.6.28, $G(-a_n) \leq G(-b_n)$ should be $G(-a_n) \geq G(-b_n)$ and $G(-x) \leq G(-y)$ should similarly be $G(-x) \geq G(-y)$.
p. 131-132$D_-$ and $D_+$ should be $D^-$ and $D^+$ respectively throughout. In the second display, $m(E_{r,R})$ should be $m(E_{r,R} \cap [a,b])$. 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.
p. 133In the proof of Lemma 1.6.31, $F^+$ should be $F_+$. At the end of the proof, add that by continuity of $F_c$, one can assume that $a,b$ avoid the jumps.
p. 134On the eighth line: “$G$ is discontinuous” should be “$F$ is discontinuous”.
p. ?In the proof of Theorem 1.6.25, in the display, Lebesgue measure $m()$ is missing on the left-hand side.
p. 135Before 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, $x_{i+1}$ should be $x_{i-1}$, and similarly on p. 136, 137. “Since $F'_\varepsilon$ is almost everywhere differentiable” should be “Since $F_\varepsilon$ is almost everywhere differentiable”.
p. ?After Remark 1.6.38, “monotone non-increasing” should be “monotone non-decreasing”.
p. ?In Exercise 1.6.39, a $dx$ is missing in the integral.
p. 137In the second paragraph, “it suffices to (by writing $F = F_+-(F_+-F_-)$ to show that $F_+-F$” should be “it suffices (by writing $F = F^+ - (F^+-F)$) to show that $F^+-F$“. “is a monotone increasing function” -> “is a monotone non-decreasing function”, and “for all $a \leq b$” should be stated after the second display.
p. 141in the definition (i) after Remark 1.6.38, “contains $x_0$” should be “whose closure contains $x_0$“.
p. 144In 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 $t_* \in [t,b_n)$“.
p. ?In Exercise 1.6.47, the last two parts of the exercise should be numbered (vii) and (viii) rather than (1) and (2).
p. 145bottom: “$f'(x)$ exists” should be “$F'(x)$ exists”. After Exercise 1.6.52, “ensure the almost everywhere existence” should be “ensure the absolute integrability of the derivative”.
p. 149-152In Section 1.7.1, “Caratheodory extension theorem” should be “Caratheodory lemma” throughout.
p. 150Exercise 1.7.2: “Lebesgue outer measurable” should be ” the Lebesgue outer measure”
p. 151In the last two displays, and in the first display on the next page, $E_{N+1} \backslash \bigcup_{n=1}^N E_n$ may be simplified to $E_{N+1}$. In the second paragraph, “a disjoint sequence of” should be “a sequence of disjoint”.
p. ?In Remark 1.7.6, ${\mathbf R}^n$ should be ${\mathbf R}^d$.
p. ?In Example 1.7.8, “finite unions” should be “finite boolean combinations”.
p. 156In Theorem 1.7.9, $-\infty < b < a < \infty$ should be $-\infty < a < b < \infty$. In the second paragraph of the proof of this theorem, before “, adopting the obvious conventions”, add “to be the required value of ${\mu_F(I)}$ given by (1.33) (e.g., ${|[a,b]|_F = F_+(b)-F_-(a)}$)”. In the statement concerning disjoint intervals that share a common endpoint, add “in such a fashion that their union is also an interval”.
p. 157Before (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 $\inf_{U \supset E_n} \mu_0(E_n)$. “We suppose that $E = {\mathcal B}_0$” should be “We suppose that $E \in {\mathcal B}_0$“.
p. 159In Exercise 1.7.13, add a right parenthesis after “absolutely integrable”.
p. 160In Exercise 1.7.14(ii), “delta functions” should be “Dirac measures” for consistency. In the first line, $-\infty < b < a < \infty$ should be $-\infty < a < b < \infty$.
p. 161In Exercise 1.7.18 (i), $Y \in B_Y$ should be $F \in B_Y$. In the second display, $\{ \pi_Y^{-1}(E): E \in {\mathcal B}_Y \}$ should be $\{ \pi_Y^{-1}(F): F \in {\mathcal B}_Y \}$
p. 162Exercise 1.7.19(ii) is not correct as stated and should be deleted.
p. 163In the third paragraph of the proof of Proposition 1.7.11, $\sum_{n=1}^\infty \mu(S_n)$ should be $\sum_{ n=1}^\infty \mu_0(S_n)$.
p. 164“integrate in $X$” should be “integrate in $x$“.
p. 165Exercise 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”. In Example 1.7.13, $({\mathbf R}^d, {\mathcal L}[{\mathbf R}^{d'}])$ should be $({\mathbf R}^{d'}, {\mathcal L}[{\mathbf R}^{d'}])$. In Exercise 1.7.21, “$\sigma$-finite sets” should be “$\sigma$-finite measure spaces”.
p. 167The sentence preceding Theorem 1.7.18 should be deleted. In parts (i) and (ii) of that theorem, the integrals should be indicated to be over the $y$ and $x$ variables respectively. In Remark 1.7.16, “Tonelli’s theorem for sums” should be “Tonelli’s theorem for series”.
p. 168in (1.37), the third integral should have X and Y interchanged (as well as the measures $d\mu_Y(y)$ and $d\mu_X(x)$). In the proof of Theorem 1.7.18, Exercise 1.4.28 should be Exercise 1.4.26.
p. 169In Exercise 1.7.22, “the counting measure (…) $\#$” should be “the counting measure $\#$ (…)”, and “Let … is … and … is ” should be “Let … be … and … be”. The hint in (iv) should be refined: one of the two integration orders can be used to build a candidate measure, but the other one is problematic. Instead, one should use Caratheodory’s theorem with a suitable premeasure. In the second line of (1.38), the integral should be over $Y$ rather than $X$, and $d\mu_X(x)$ should be $d\mu_Y(y)$. In the seventh line from the bottom, “equal to one for every $y$” should be “equal to one for every $x$“.
p. 170In 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 $\int_Y (\int_X f(x,y)\ d\mu_X(x))\ d\mu_Y(y)$. In Remark 1.7.22, “$\sigma$-finite setting” should be “non-$\sigma$-finite setting”.
p. 171In Exercise 1.7.24, “Show that if” should be “Show that”.
p. 175In the last complete paragraph, “for thus purpose” should be “for this purpose”.
p. 186In Section 2.1.18, the introduction of the limit inferior should use := instead of =.
p. 187In the display before Remark 2.2.3, $h \to {\bf R}^d \backslash \{0\}$ should be $h \in {\bf R}^d \backslash \{0\}$.
p. 188After (2.2), $\frac{\partial f}{\partial x_0} f$ should just be $\frac{\partial f}{\partial x_0}$.
p. ?In the proof of Theorem 2.2.4, “as ${\bf Q}^d$ is rational” should be “as ${\bf Q}^d$ is countable”.
p. 189In the second paragraph, a comma is missing between “For $v=0$” and “$E_v$ is clearly”. In the third paragraph, “$E$ is a null set” should be “$E_v$ is a null set”. In the fourth paragraph, “${\bf Q}^d$ is rational” should be “${\bf Q}^d$ is countable”. In the definition of $E^{y_0}$, $E$ should be $E_v$. The definition of F-volume is missing the text “to equal the right-hand side of (1.33)”; also, one could replace “F-volume” by “F-length” to be more descriptive.
p. 191The modified function $F$ should take values in ${\bf R}$ rather than ${\bf R}^d$.
p. 194In the final sentence of Section 2.3, ${\bf E} X$ should be ${\bf E}(X)$ for notational consistency.
p. 195In Exercise 2.4.1(3), $E$ should be $E_A$. In Exercise 2.4.1(8), $x_B \to f(x_B,x_{A\backslash B})$ should be $x_B \mapsto f(x_B,x_{A\backslash B})$.
p. 197In 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 $\sum_{N'=1}^N \varepsilon/2^{N'+1}$, and the final $\varepsilon - \varepsilon/2^N$ should just be $\varepsilon/2$. In the seventh paragraph, $\bigcup_{n=1}^N E_N$ should be $\bigcup_{n=1}^N E_n$.
p. 205The 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)”.
Contributors
Thanks to all those who have contributed corrections. Corrections received on or before 2026-07-09 were reported by the readers listed below over the years; individual per-erratum attributions for these legacy entries were not preserved in migration.