ERRATA TO “REAL ANALYSIS,” 2nd edition

(6th and later printings)

G. B. Folland
Last updated February 23, 2024.

Additional corrections will be gratefully received at folland@math.washington.edu .

Page 7, line 12: Y ∪ {y0 } → B ∪ {y0 }

Page 7, line −12: X ∈ → x ∈
Page 8, next-to-last line of proof of Proposition 0.10: E → X
Page 12, line 17: a ∈ R → x ∈ R (two places)
Page 14, line 16: x ∈ X → x ∈ X1
Page 14, line 17: whenver → whenever
Page 22, line 2: susbset → subset
Page 24, Exercise 1, line 1: A family → A nonempty family
Page 24, Exercise 3a: disjoint → disjoint nonempty
Page 29, Proposition 1.10: The hypothesis that X ∈ E was included only to guarantee
that µ∗ (A) is well-defined for all A ⊂ X, and with the understanding that inf(∅) = +∞,
it isSunnecessary.
P ∗ The proof extends to the general case without change, as the condition
∗ ∗
µ ( Aj ) ≤ µ (Aj ) is nontrivial only when µ (Aj ) < ∞ for all j.
Page 34, line 1: n1 Jj →
S Sm
1 Jj
Page 35, line −3: open h-intervals → open intervals
Page 37, line -1: countable → countable set.
Page 38, line −4: ∞
P P∞
0 → 1
Page 40, line 2 of §1.6: 2.7 → 2.8
Page 45, line 5: [∞, ∞] → [−∞, ∞]
Page 45, line 8: 2.3 → 1.2
Page 47, Figure 2.1: The graph of ϕ1 should have an extra “step” where the ordinate goes
from 1 to 23 and then from 23 to 2, rather than directly from 1 to 2.
Page 49, line −8: inegrals → integrals
Page 56, last line of proof of Theorem 2.27: (x, t) → (x, t0 )
Page 60, Exercise 27c: log(b/a) → log(a/b)
Page 60, Exercise 31e: s2 → a2
Page 61, line 9: repectively → respectively
Page 66, line −4: ∞ → E= ∞
T T
1 En 1 En
R R y
Page 67, next-to-last line of Theorem 2.37: f y dν → f dµ.

1
Page 69, Exercise 49a: M × N → M ⊗ N
Page 69, Exercise 50: Either assume f < ∞ everywhere or use the condition y < f (x) to
define Gf . Also, M × BR → M ⊗ BR .
Page 70, proof of Theorem 2.40, line 2: rectangles → rectangles, which may be assumed
bounded,
Page 72, line 5: definitons → definitions
P P
Page 75, line 9: j (xj − aj )(∂g/∂xj )(y) → k (xk − ak )(∂gj /∂xk )(y)
Page 75, line 9: joning → joining
Page 76, line 6: ∞
S T∞
1 Uj → 1 Uj
Page 76, line −7: f ◦ G → f ◦ G| det DG|
Page 76, line −5: G(Ω)) → G(Ω)
R R∞
Page 77, Exercise 58: → 0
Page 79, line 2: (a, b] × E → (a, b] × E or (a, ∞) × E
P P
Page 87, line 3: ν(Aj ) > → ν(Aj ) ≥
R
Page 88, Exercise 3c: |f | ≤ 1 → |f | ≤ 1 and E f dν exists [The latter condition is
automatic when ν is finite.]
R R
Page 88, Exercise 6: f dµ → E
f dµ
Page 90, line −6: f → fj
Page 102: (3.24) should be interpreted as “TF (b) = TF (a) + sup{. . . }” in the case TF (b) =
TF (a) = ∞.
Page 103, line −5: ± 12 F (−∞) → ∓ 12 F (−∞)
Page 104, line 7 of proof of Lemma 3.28: x0 < · · · → x = x0 < · · ·
Page 104, line −12: n1 →
P Pm
1
Page 105, line 2 of proof of Proposition 3.32: suppose that E → suppose that F is
absolutely continuous and E
Page 105, line 5 of proof of Proposition 3.32: µ(Uj ) < δ → m(Uj ) < δ
Page 106, line 4: greatest integer less than δ −1 (b − a) + 1 → smallest integer greater
than δ −1 (b − a)
Page 107, Exercise 28b: µTF (E) → µTF (E)
Page 115, line −12: Propostiion → Proposition
Page 120, line −2: a neighborhood → an open neighborhood
Page 125, line 16: is a set → is a nonempty set
Page 144, line 12: an LCH → a noncompact LCH
Page 145, paragraph after the end-of-proof sign, line 3: locally compact → locally
compact and noncompact
Page 146, Exercise 73: In the definition of completely regular algebra, add the condition that
the algebra be closed under complex conjugation. Also, in parts (a), (b), and (d), the word

2
“Hausdorff” is redundant since it is incorporated in the definition of “compactification” on
p. 144.
Page 146, Exercise 73c: contains F → contains F and the constant functions
Page 146, Exercise 73d: Insert “(up to homeomorphisms)” after “of X”.
Page 159, next-to-last line of proof of Theorem 5.8: Moroever → Moreover
Page 165, line 6: x ∈ X → x ∈ X
Page 166, line −2 of proof of Theorem 5.14: (1 − t)x + (1 − t)z → (1 − t)x − (1 − t)z
Page 166, line −1: Uxαj ϵj → U0αj ϵj
Page 167, line 3: pαj (y) < ϵ → pαj (y) ≤ ϵ
Page 167, bulleted item at bottom (continuing to next page): CX should be replaced by the
space of locally bounded functions on X, i.e., the space of all complex-valued functions f on
X such that pK (f ) < ∞ for all K.
Page 174, line 2: paralellogram → parallelogram
Page 174, lines −8 and −4: X → H
Page 177, line 1: eα → uα and X → H
Page 179, next-to-last line of notes for §5.1: coincides with → extends
Page 179, line −2: x1 ∈ X0 → x0 ∈ X0
Page 194, line −3, “simple consequence”: Actually, all the y-sections of the set {(x, y) :
|f (x, y)| > ∥f (·, y)∥∞ } have µ-measure 0, and you need Tonelli to deduce that µ-almost all
the x-sections have ν-measure 0.
Page 196, Exercise 28b: J1 f → Jα f (Hint: Focus on the behavior of Jα f (x) as
x → ∞.)
Page 197, line −2: on (0, ∞), → on [0, ∞) such that ϕ(0) = 0,
Page 203, statement of Marcinkiewicz interpolation theorem, last sentence: If p0 = p1 (so p
doesn’t vary), Bp and |p − pj | should be replaced by Bq and |q − qj |.
p q
Page 204, last line of (6.33): Cj j → Cj j
Page 206, Theorem 6.36, line 4: 1 ≤ p < ∞ → 1 ≤ p < q/(q − 1)
Page 208, Exercise 41: For the case p = ∞, assume µ semifinite.
Page 208, Exercise 45, lines 3 and 4: T is weak type (1, nα−1 ) and strong type (p, r) where
1 < p < n(n − α)−1 and r−1 = p−1 − (n − α)n−1 .
Page 210, final sentence: Theorem 6.36 was discovered independently, a little earlier than
[51], by D. R. Adams (A trace inequality for generalized potentials, Studia Math. 48 (1973),
99–105).
Page 212, line 13: a Borel measure → a Borel measure that is finite on compact sets
Page 217, lines 7 and 8: f → f1
Page 218, line −5: χu → χU
Page 221, Exercise 15e: E ⊂ BΩ∗ → E ∈ BΩ∗

3
Page 224, line 8: Insert minus signs before the two middle integrals.
R R
Page 224, line 9: fn dµ → f dµn
Page 224, line −4 of proof of Proposition 7.19: (−∞, N ] → (−∞, −N ]
Page 224, Exercise 18, line 1: M(X) → M (X)
R
Page 225, Exercise 24b: f dµ → 0
Page 225, Exercise 24c: F (x) → 0
Page 225, Exercise 27: k functionals → k bounded functionals
Page 226, proof of Theorem 7.20, next-to-last line: π1 (K) × π2 (K) → πX (K) × πY (K)
Page 226, proof of Theorem 7.20, last line: = → ≤
Page 226, line 2 of Proposition 7.21: X ⊗ Y → X × Y
Page 226, line −2: U × V → U × V
Page 227, 4th and 3rd lines before Lemma 7.23: Replace the clause “Exercises 12 and . . .
µ×ν”
b by “Exercise 12 shows that µ×ν({0}
b × R) ̸= 0 = µ × ν({0} × R)”. (The semifinite
part of µ×ν disagrees with µ × ν on {(x, x) : x ∈ [0, 1]}; see Exercise 2.46.)
b
Page 228, line 3: m
T Tn
1 → 1
Page 229, line −10: BX × BY → BX ⊗ BY
Page 232, line 5 of paragraph 3: L1 (µ) → L1 (µ)∗
Page 242, line 12: ∥g∥(N +n+1,α) → ∥g∥(N +n+1,0)
Page 246, Exercise 9: Assume p < ∞.
Page 247, line 2 of Theorem 8.19: Tn → Zn
P P
Page 250, line −2: |γ|≤|β| ∥f ∥(N +n+1,γ) → |γ|≤N ∥f ∥(|β|+n+1,γ)
2 2
Page 251, line 4: −2πae−πax → −2πaxe−πax
Page 254, line 5: ZN → Zn
Page 254, line 4 of proof of Theorem 8.32: 8.35 → 8.31
Page 255, Exercise 16a: ∥f ∥u → ∥fk ∥u
Page 256, line 1: right → left
Page 257, first paragraph of §8.4: To conclude that f = g, one needs the injectivity of
the Fourier transform on L1 (Tn ). There are several ways to establish this without invoking
Fourier inversion, e.g.: (1) Show that if fb = 0 then f defines the zero functional on C(Tn ).
(2) Use Exercise 28 (p. 262); from part (a), if fb = 0 then Ar f = 0, and from part (b), the
mass of Pr concentrates at the origin as r → 1, whence ∥Ar f − f ∥1 → 0.
Page 259, line 9: f2 ∗ ϕt (ξ) → f2 ∗ ϕt (x)
P
Page 259, line 3 of proof of Theorem 8.36: The sum on the right should be κ∈Zn .
Page 261, line 7: e−2πiκx → e2πiκx
Page 264, line 4: e2π(2m+1)x → e2πi(2m+1)x
Page 268, formula (8.46): 21 − x − [x] → 21 − x + [x]

4
Page 269, line 6: Sm (aj ) → Sm f (aj )
Page 269, Exercise 35a: ϕ → ϕm (two places)
Page 272, Exercise 39: On line 2, positive → nonnegative. Also, replace line 3 by the
following: at m α α+1 α+m−1
, m , m for some α ∈ [0, 1) and m ∈ N, in which case µ b(jm) = e−2πijα
for all j ∈ Z.
Page 273, line 7: if for all → for all
Page 274, line −1: (t2 + |x|2 )−(n+1)/2 → (t2 + |x|2 )(n+1)/2
Page 276, Exercise 43: e−|x|/2 → 21 e−|x|
Page 286, line 3: ϕ(y) → ϕ(x)
Page 286, lines −13 and −5, and page 287, lines 1 and 3: U → V
Page 288, line −10: ψ(ϵx) → ψ(x/ϵ)
Page 289, Exercise 7, line 2: f agrees → there exists a constant c such that f + c agrees
Page 291, Exercise 13: f ∗ ψt → F ∗ ψt
Page 293, line −2: (1 + |x|)N → (1 + |x|)−N
Page 293, line −1: ∥ϕ∥(0,N ) → ∥ϕ∥(N,0)
Page 294, line 3: by (ii) → by the preceding example
Page 296, line −9: xj → ξj
Page 297, line 7: One → On
Page 297, proof of Proposition 9.14, line 3: f = gb → f = g ∨
Page 297, line −3: fb(κ) → Fb(κ)
Page 300, Exercise 28, line 2: |ξ|α−2 → |x|α−2
Page 303, lines 5–6: Fourier transform is gb(ξ) → inverse Fourier transform is g ∨ (ξ)
Page 303, line 7: (1 + |ξ|2 )s → (1 + |ξ|2 )−s
Page 309, Exercise 34c: Λa → Λα . Also, apologies for the two conflicting uses of the
letter α; one might prefer to replace ∂ α and |α| by ∂ β and |β|.
Page 320, line −1: the the → the
Page 323, line 5: lim sup n−1 |Sn | < ϵ → lim sup n−1 |Sn | ≤ ϵ
Page 325, Exercise 17, line 2: smaple → sample
Page 325, Exercise 17, line 9: Xj − Mj → Xj − Mn
Page 325, line 3 of §10.3: e(t−µ) /2σ
2 2 2 2
→ e−(t−µ) /2σ
Page 326, line −6: Xn → Xj
Page 331, line −7: exp(· · · ) → exp(− · · · )
Page 332, formula (10.23): exp(· · · ) → exp(− · · · )
Page 341, proof of Proposition 11.3, line 3: it → if
Page 344, proof of Theorem 11.9, end of line 2: Delete “h ∈ Cc+ and”.
Page 348, Exercise 9c: In general it is not µ that is decomposable but rather its extension µ
to the σ-algebra of µ∗ -measurable sets as explained on p. 215.

5
Page 349, line 3: µ∗ (A) ∪ µ∗ (B) → µ∗ (A) + µ∗ (B)
Page 349, line −11: B 2k−3 → B2k−3
Page 349, line −7: ∞
P P∞
n+1 → n
Page 350, proof of Proposition 11.17: Concerning the applicability of Proposition 1.10, see
the correction to Page 29 in this errata list.
Page 357, Figure 11.1(b): In the bottom figure, the small triangle in the center should not
be shaded.
Page 358, line 10: C( X) → C(X)
Page 358, line −7: xi1 ···xk → xi1 ···ik
∂yi ∂yj ∂yk ∂yl
Page 362, first display: →
∂xk ∂xl ∂xi ∂xj
Page 373, reference 131: of → in
Page 373, reference 139: in → on
Page 378, line −2: CS ′ → S′