ERRATA and IMPROVEMENTS

The columns titled Aut-hor contain abbreviations of the names of the people who have graciously pointed out the error following. They are:

RB ▷ Robert B. Burckel,  burckel@math.ksu.edu
MR ▷ Mike Revesz,  revesz@math.utexas.edu
SB ▷ Sterling Berberian,  berb@math.utexas.edu
PC ▷ Piotr T. Chrusciel,  Piotr@gargan.Math.Univ-Tours.fr
MM ▷ Marti Llaurado Magrinya,  mllm1@wanadoo.es

Please send errors you found or suggestions to:

KB ▷ Klaus Bichteler,  kbi@math.utexas.edu

Many of the errata pointed out came with suggested changes, for which I thank their authors. If a change below contains yet more mistakes, they are mine; I may not have followed the suggestion or may have made up a change of my own. To ease reading the tables, plain falsehoods and errata that obscure the mathematics start in boldface and are highlighted like this. Typos and slight improvements appear in unadorned fashion.

Chapter I

Pa- Li- Au- ◁ Old  ▷ New  Com
ge ne thor◁ Text  ▷ Text  ment
6 -11 RB ◁ of functions  ▷ of real valued functions
8 -12 PC ◁ dense in E  ▷ dense in E in the topology of uniform convergence

9 3–4 RB ◁ ℝ^{n}  ▷ ℝ^{m}  (avoiding two meanings of n)

9 19 RB ◁ to ϕ in the sense ▷ to \overline{ϕ} in the sense
9 -9 RB ◁ is then  ▷ is, then
10 -6 RB ◁ and t^{2} − ϕ_{k} ≤ ϵ^{2} … ▷ and t^{2} − ϕ_{k}(t) ≤ ϵ^{2} …
10 -4 RB ◁ ((k − 1)ϵ, kϵ)  ▷ ((k − 1)ϵ, (k + 1)ϵ)
13 1 RB ◁ vanish then  ▷ vanish, then
13 3 RB ◁ point z ∈ S then ▷ point z ∈ S, then
13 -15 RB ◁ {[ϕ_{s_{i},t} : 1 ≤ i ≤ n} ▷ {ϕ_{s_{i},t} : 1 ≤ i ≤ n}  extraneous [
13 -3 RB ◁ interval S,  ▷ interval S =^{def} [−M, M]
15 -4 MR ◁ pseudometric  ▷ pseudometric
166 13 RB ◁ For ϕ ∈ E_{+}  ▷ For ϕ ∈ E
166 17 RB ◁ vanishes on ϕ ≥ 1 ▷ vanishes on [ϕ < 1]
166 17 RB ◁ [(2ϕ − 1)_{+} ≥ 1]  ▷ [(2ϕ − 1)_{+} < 1]
166 23 RB ◁ αs^{α−1} ∧ δ^{(α−1)/α} ▷ (αs^{α−1}) ∧ δ^{(α−1)/α} 1
166 -3 RB ◁ r_{i}·e^{α_{i}t}  ▷ r_{i}·e^{−α_{i}t}
167 2 RB ◁ e^{αt}  ▷ e^{−αt}
167 11 RB ◁ after I.3.28 insert: We treat the case that E is a ring. 2
20 15 RB ◁ A collections  ▷ A collection
20 -16 RB ◁ left–continuous,  ▷ left–continuous with right limits, 3
20 -6 RB ◁ are  ▷ is
21 -16 RB ◁ _{∞} < ^{∫ ♮}f  ▷ −∞ < ^{∫ ♮}f
166 -3 MR ◁ everyone  ▷ every one
26 -1 RB ◁ continous ▷ continuous also on p.27 line 8
27 -7 RB ◁ ∑_{n} b_{n} − a_{n} < ϵ  ▷ ∑_{n} (b_{n} − a_{n}) < ϵ

Chapter I Continued

Pa- Li- Au- ◁ Old  ▷ New  Com
ge ne thor◁ Text  ▷ Text  ment
27 -4 RB ◁ satisfies  ▷ possesses

27 -0 RB ◁ insert exercise 6.17  4

169 3 RB ◁ for  ▷ For

169 -20 KB ◁ replace the text from (ii) Let . . . to line -13 by:

(ii): The functions ϕ_{n} =^{def} ∑_{i=1}^{n} 2^{−i}(i−1, i] ∈ E[ℝ] form a ∥ ∥^{♮}–Cauchy sequence. Assume f is a ∥ ∥^{♮} –norm limit. Then f cannot vanish outside (−M, M], whatever M ∈ ℕ: if it did then ψ_{n} =^{def} ϕ_{n}·(−M, M] ⟶_{n→∞} f·(−M, M] = f in ∥ ∥^{♮}–norm and ∥ϕ_{n} − ψ_{n}∥^{♮} → 0; yet ∫ ϕ_{n} − ψ_{n} ≥ 2^{−M} for n > M: f ∉ F^{♮}, so it is not Riemann integrable, a contradiction. Thus (ϕ_{n}) has no ∥ ∥^{♮} –limit.

1 Actually, it does not matter for the argument whether the old text is read as (αs^{α−1}) ∧ δ^{(α−1)/α} (intended) or as α(s^{α−1} ∧ δ^{(α−1)/α}). The same change should be made on page 23, lines -13, -10, -9.

2 The point of this change is to make sure that the “∈” of “Then ϕ_{n} =^{def} ϕ′_{n} · σ · ρ_{n} ∈ E ” in line 17 is justified. The exercise is rather simple if E is a self–confining vector lattice closed under chopping: let E ∋ ϕ_{n} → \overline{ϕ} ∈ \overline{E}_{00} uniformly. There exist ψ ∈ E = E_{00} confining \overline{ϕ}, without loss of generality ψ > sup_{n} ∥ϕ_{n}∥_{∞} on [\overline{ϕ} ≠ 0]. Then (−ψ) ∨ ϕ_{n} ∧ ψ belongs to E, is confined by any function confining ψ, and converges uniformly to \overline{ϕ}.

3 Robert B. Burckel gives the counterexample ϕ = ∑_{n∈ℕ} 1_{( \frac{1}{2n+1} , \frac{1}{2n}]} to the “if” part of the claim as stated.

 4 Insert:

exercise 6.17 Show that if a function f ∈ F^{♮} has ∥f∥^{♮} = 0, then the set [f ≠ 0] is Lebesgue–negligible.

(Suggested by Robert B. Burckel)

Chapter II

Pa- Li- Au- ◁ Old  ▷ New  Com
ge ne thor◁ Text  ▷ Text  ment
32 12 RB ◁ stays  ▷ persists
32 13 RB ◁ (a)  ▷ (i)
32 14 RB ◁ The theorem  ▷ The putative theorem
32 16 RB ◁ (b)  ▷ (ii)
34 3 RB ◁ decreasing  ▷ positive
34 -7 RB ◁ Equation (1.1)  ▷ Inclusion (1.1)
34 -2 RB ◁ ϵ·(M + ∥ψ_{1}∥_{u})  ▷ ϵ·(2M + ∥ψ_{1}∥_{u})
37 19 RB ◁ K ⊂ S compact  ▷ K a compact subset of S
37 F^{2} RB ◁ point set topology  ▷ point–set topology
171 5 RB ◁ C_{n}^{1} = C_{n}^{2} = C  ▷ C_{n}^{1} ⊆ C_{n}^{2}
39 17 RB ◁ sets in R  ▷ sets in R, each contained in one of the A_{i},
41 33 RB ◁ ≤ ∥m_{1} ∥ · ∥m_{1} ∥  ▷ ≤ ∥m_{1} ∥ · ∥m_{2} ∥
42 -8 RB ◁ take the value ∞  ▷ take the value +∞
43 -12 RB ◁ {ϕ_{1,k} + ϕ_{2,k} : k ∈ ℕ} ⊂ E ▷ {ϕ_{1,k} + ϕ_{2,l} : k, l ∈ ℕ} ⊂ E
44 1 RB ◁ By Lemma 1.1  ▷ Due to the σ–additivity of ∫ (Lemma 1.1)
44 2 RB ◁ lim_{n→∞} ∫ ϕ̂_{n} > r .  ▷ lim_{n→∞} ∫ ϕ̂_{n} = ∫ ϕ > r .
44 8 RB ◁ ∫^{∗}h + h′ … ∫^{∗}ϕ_{n} + ϕ′_{n} ▷ ∫^{∗}(h + h′) … ∫^{∗}(ϕ_{n} + ϕ′_{n})

Chapter II Continued

Pa- Li- Au- ◁ Old  ▷ New  Com
ge ne thor◁ Text  ▷ Text  ment
45 10 KB ◁ p.19  ▷ page 19
45 15 RB ◁ of functions  ▷ of real–valued functions
45 19 RB ◁ realized that  ▷ realized in Lemma 3.1 that the σ–additivity

established in Lemma 1.1

45 28 RB ◁ seminorm  ▷ functional
45 31 RB ◁ (f_{n})  ▷ (f_{n}) of real–valued functions ∞
47 12 RB ◁ ring ring  ▷ ring
47 12 MR ◁ function h  ▷ function h ≥ 0
47 16 RB ◁ add: [Hint: Reduce to the case that h has compact support and consider

h_{n}(x) =^{def} inf{h(y) + nd(x, y) : y ∈ S}]

47 -12 RB ◁ extend the line:  ▷ (Use equation I (4.3) to define ∫^{♮} on all functions,

keeping in mind the convention inf ∅ = +∞ (page 4).)

47 -5 RB ◁ all positive numerical  ▷ all numerical
49 1  KB ◁ A function  ▷ A numerical function

49 -2 RB ◁ Replace lines -2,-1 by                             ∞

(iii): Let A =^{def} [f′ = f′′]. Then ∥f′∥^{∗} = ∥Af′ + A^{c}f′∥^{∗} ≤ ∥Af′∥^{∗} + ∥A^{c}f′∥^{∗} = ∥Af′∥^{∗} + 0 ≤ ∥f′∥^{∗},

whence ∥f′∥^{∗} = ∥Af′∥^{∗} = ∥Af′′∥^{∗} = ∥f′′∥^{∗}.

50 -11 MR ◁ defined.  ▷ defined, and how to read f ≤ g a.e. etc. when

f, g are merely a.e. defined.

51 9 RB ◁ ≤ 2^{−k} ▷ ≤ 2^{−k−1}
51 -10 RB ◁ ∥G · ∑_{k=K}^{∞} f′_{n_{k+1}}−f′_{n_{k}} ∥^{∗} ▷ ∥G · ∑_{k=K}^{∞} (f′_{n_{k+1}}−f′_{n_{k}})∥^{∗}
51 -2 RB ◁ f_{n_{k}} ⟶_{n→∞} f ▷ f_{n_{k}} ⟶_{k→∞} f n → k
53 15 RB ◁ ∥ϕ_{n}∥^{λ} = 2^{−n} → 0  ▷ ∥ϕ_{n}∥^{λ} = 2^{−m} < 2/n → 0
53 -9 RB ◁ . This  ▷ . This is
54 13 RB ◁ sup_{K} ∥ ∑_{k=1}^{K} f_{n_{k+1}}−f_{n_{k}}∥^{∗} ▷ sup_{K} ∥ ∑_{k=1}^{K} (f_{n_{k+1}}−f_{n_{k}})∥^{∗}
54 15 RB ◁ (5.3) (iii)  ▷ (5.3) (ii)
55 6  RB ◁ Theorem 6.1  ▷ Theorem I.6.3
55 8  RB ◁ permance  ▷ permanence
55 16 RB ◁ (f ∧ 1)_{−}  ▷ −(f ∧ 1)_{−}
55 -5 MR ◁ and all  ▷ and for all
172 -19 RB ◁ ϕ_{k} → h  ▷ ψ_{k} → h
172 -19 RB ◁ ∥h∥^{∗} ≥ sup ∥ψ_{k}∥^{∗} = ∞  ▷ ∥h − ψ_{1}∥^{∗} ≥ sup_{k} ∥ψ_{k} − ψ_{1}∥^{∗} = ∞
172 -10 RB ◁ lim_{n} ∥g_{N}∥^{∗} = lim inf_{n} ∥f_{n}∥^{∗} ▷ lim_{N} ∥g_{N}∥^{∗} ≤ lim inf_{n} ∥f_{n}∥^{∗} n → N
172 -2 RB ◁ mesurability  ▷ measurability
56 -9 RB ◁ Extend: L^{1} is a module over \overline{E} and over L_{b}^{1} .
56 -6 RB ◁ Delete Exercise 5.12 ▷ 1
58 -7 KB ◁ | ∫ ϕ_{n} − ϕ′_{n}|  ▷ | ∫(ϕ_{n} − ϕ′_{n})|
59 3  RB ◁ ∫ |f| = ∫^{∗}|f| = ∥f∥^{∗}  ▷ ∫ |f| = ∥f∥^{∗}  delete:∫^{∗}|f|

59 6  RB&MM ◁ the whole item should read as follows:         2

(iv) Suppose the sequence (f_{n}) of integrable functions converges ∥ ∥^{∗}–a.e. to f and ∥ sup |f_{n}|∥^{∗} is finite. Then f_{n} → f in mean, f is integrable, and ∫ f_{n} → ∫ f.

59 -10 RB ◁ ∫ |f| = lim ∫ ϕ_{n}.  ▷ ∫ |f| = lim ∫ |ϕ_{n}|.
59 -6–-5 RB ◁ (iii) …  ▷ (iii) and (iv) are evident from Theorem 5.6.

Chapter II Continued

Pa- Li- Au- ◁ Old  ▷ New  Com
ge ne thor◁ Text  ▷ Text  ment

59 -2 KB ◁ Replace this line by:                         ∞

Assume E is a vector lattice closed under chopping as well as a ring. (i) Define, in the manner of Definition 4.4, ∫^{∗} and ∫_{∗} for numerical functions that are merely ∥ ∥^{∗}–a.e. defined, and show that the properties listed in Proposition 3.2 and Exercise 3.14 persist. (ii) Then show that an a.e. defined numerical function f is (E, ∥ ∥^{∗})–integrable if and only if

60 1 KB ◁ Delete: (see Exercise 3.14)
60 2 RB ◁ Insert: Exercise 6.2’ Assume that ∫ = ∥ ∥^{∗} on E_{+} . If (f_{n}) is an increasing or decrea-

sing sequence of integrable functions and the sequence ( ∫ f_{n}) is bounded then the pointwise limit f =^{def} lim f_{n} is integrable and f_{n} → f in mean. [Hint: Apply 6.1 (ii) and the MCT to (f_{n} − f_{1}) or (f_{1} − f_{n}).]

173 6 RB ◁ after II.6.2 insert the following lines:         ∞

(i): Set f̌ = f̂ = f on the set D where f is defined, f̌ = −∞ and f̂ = +∞ on the negligible set D^{c}. If ∫^{∗}f̌ < s then there is an h ∈ E^{↑} with f̌ ≤ h and r =^{def} ∫^{∗}h < s. As f̂− f̌ = ∞·D^{c} is negligible, there is an h_{0} ∈ E^{↑} with f̂− f̌ ≤ h_{0} and ∫^{∗}h_{0} < s − r. Then E^{↑} ∋ h + h_{0} > f̂ and ∫^{∗}h + h_{0} < s. Thus ∫^{∗}f̂ ≤ ∫^{∗}f̌ and ∫^{∗}f̌ = ∫^{∗}f̂. It is thus unequivocal to define ∫^{∗}f =^{def} ∫^{∗}f′, where f′ is any function defined everywhere and equal to f on D. ∫_{∗}f is of course defined similarly. To check the properties claimed is now a matter of (a lot of) bookkeeping. [We see here that upper and lower integrals are rather user–unfriendly and the employ of means is much preferable.] (ii): In view of (i) neither hypothesis nor conclusion change if f is replaced with a real–valued function that is everywhere defined. Continue: Assume …

60 4 RB ◁ dominated  ▷ dominated by an integrable function
60 6 RB ◁ evident,  ▷ evident from Theorem 5.6,
60 6 RB ◁ This  ▷ The italicized conclusion
60 11 RB ◁ Extend this line: mean. (The mean is of course the Daniell mean

for the measure in question, and integrability is understood with respect to that.)

173 8 RB ◁ f ≤ ϕ + h_{n}  ▷ f ≤ ϕ_{n} + h_{n}
173 20 RB ◁ and II.6.7  ▷ II.6.2 (i), and II.6.7 below ∞
173 22 RB ◁ 2g_{n}  ▷ g + g_{n}
173 22 RB ◁ Replace Therefore with

By applying the preceding conclusion to |f − f_{n}| in the role of f_{n} and g + g_{n} in the role of g_{n}, it follows that

173 -14 RB ◁ with  ▷ with h_{n} ≥ f and
173 -12 RB ◁ (MCT) ▷ (Exercise 6.2’) see p60 l2
173 -11 RB ◁ lim_{N→∞} ⋀_{n=1}^{N} h_{n}.  ▷ lim_{N→∞} ⋀_{n=1}^{N} h_{n}, which is integrable by

Theorem 5.5 and the MCT (page 54).

173 -7 RB ◁ ∫^{∗} is subadditive on F^{∗}, but not in general. Replace answer II.6.7 by:

Assume f is the integrable function. If ∫^{∗}(f + f′) < r ∈ ℝ then there is a function h ∈ E^{↑} exceeding f + f′ and having upper integral ∫^{∗}h < r. By 6.2’, h is integrable. Then f′ ≤ h − f ∈ L^{1} and consequently ∫^{∗}f′ ≤ ∫^{∗}(h − f) = ∫(h − f) = ∫ h − ∫ f, which implies ∫^{∗}f′ + ∫^{∗}f ≤ r. Thus ∫^{∗}(f + f′) ≥ ∫^{∗}f + ∫^{∗}f′. From this ∫^{∗}f′ = ∫^{∗}[(f + f′) + (−f)] ≥ ∫^{∗}(f + f′) + ∫^{∗}(−f) = ∫^{∗}(f + f′) + ∫ (−f) = ∫^{∗}(f + f′) − ∫^{∗}f, which is the reverse inequality.

61 3 RB ◁ S  ▷ ℝ

61 4 RB ◁ Delete exercise 6.9 ▷  3
173 -1 RB ◁ and delete this answer of it

61 -10 MR ◁ A set  ▷ a set
61 -9 MR ◁ are immediate  ▷ are thus immediate
62 20 MR ◁ the el…σ–finite  ▷ the el…σ–finite
62 -18 RB ◁ countable many  ▷ countably many
62 -13 MR ◁ integrands whose  ▷ integrands in E whose
62 -12 MR ◁ mean is  ▷ mean on E is

Chapter II Continued

Pa- Li- Au- ◁ Old  ▷ New  Com
ge ne thor◁ Text  ▷ Text  ment
174 8 RB ◁ sets covers  ▷ sets (see Proposition 7.5) covers

64 4 RB ◁ A_{n}  ▷ A_{n,k}  also on lines 5,11

64 8 RB ◁ Rieman  ▷ Riemann
64 17 RB ◁ ∫ s_{n}  ▷ ∥s_{n}∥^{∗}  also on line 19
66 1 RB ◁ her  ▷ his  silly, yes
174 24 RB ◁ f  ▷ 1_{A}
174 27 RB ◁ U = ⋃_{i,n=1}^{∞}(a_{n}^{i}, b_{n}^{i} + ϵ2^{−n−i−1}) ▷ U = ⋃_{n=1}^{∞} ⋃_{i=1}^{I(n)}(a_{n}^{i}, b_{n}^{i} + ϵ2^{−n−i−1})
174 -9 RB ◁ … B_{k} ∩ (−1, 1))  ▷ … B_{k} ∩ (−1, 1)), B_{k} as on page 67
67 -2 MR ◁ sequence f_{n}  ▷ dominated sequence (f_{n})
67 -1 MR ◁ zero  ▷ zero that is dominated by an

elementary integrand and

∞ Robert Burckel found several instances of expressions like f − g where both f and g could take the value +∞. The sign ∞ points out that this is the reason for the change made.

1 Robert Burckel gives the counterexample ℓ^{1}(ℕ) to (ii) as stated. If exercise II.5.12 is to be kept, L^{1} should be replaced by L^{1} , the assumption ∥1∥^{∗} < ∞ must be added, and the whole thing should be moved as Exercise II.3.2 on page 108.

2 The condition that lim ∫ f_{n} be finite is not sufficient, as witness the mean ∥f∥^{∗} = ∫^{∗} |f| + |f(0)|: it majorizes the Lebesgue integral, yet f_{n} =^{def} n{0} defeats the claim. One must require in addition that ∥ϕ∥^{∗} = ∫ ϕ for E ∋ ϕ ≥ 0. The deleted claim is dealt with in Remark 6.3 on the next page anyway.

3 This is patently false. Robert Burckel gives the counterexample E = {0}, L^{∗} = L^{1}(ℝ).

Chapter III

Pa- Li- Au- ◁ Old  ▷ New  Com
ge ne thor◁ Text  ▷ Text  ment
70 7 RB ◁ elementary integrands ▷ following functions from \overline{E}
70 12 RB ◁ k · ϕ_{k}  ▷ k · |ϕ_{k}|,
70 17 KB ◁ E^{↑}  ▷ \overline{E}_{+}^{↑} = E_{+}^{↑}
70 -9 RB ◁ to f  ▷ to f′
70 -8 RB ◁ U^{c} ∪ [f ≠ f′]  ▷ U^{c} ∩ [f ≠ f′]
71 -6 RB ◁ f b a  ▷ f be a
72 -13 RB ◁ Extend:An integrable function is measurable
72 -9 MR ◁ every function  ▷ every real–valued function
73 11 RB ◁ alternate  ▷ alternative
73 12 RB ◁ everywhere defined ▷ everywhere–defined
73 -9 MR ◁ refer  ▷ defer
73 -8 MR ◁ permance  ▷ permanence
73 -3 RB ◁ a function ▷ some function \overline{ϕ}_{n} also on p.74 l.16
174 -5 RB ◁ integable  ▷ integrable
73 -3 MR ◁ everyone  ▷ every one

Chapter III Continued

Pa- Li- Au- ◁ Old  ▷ New  Com
ge ne thor◁ Text  ▷ Text  ment
174 -3 MR ◁ everyone  ▷ every one  same on line -1
175 2 RB ◁ Proposition II.7.1  ▷ Propositions II.7.1 and II.7.5
175 3 RB ◁ A  ▷ A_{1}
74 15 MR ◁ everyone  ▷ every one
74 -1 RB ◁ tt Extend: If f_{n} is measurable and ∥f_{n} − f∥^{∗} ⟶_{n→∞} 0 then f is measurable.
75 9 MR ◁ and check  ▷ to check
76 11 MR ◁ conlude  ▷ conclude
77 8 MR ◁ \overline{α}^{1}, \overline{α}^{2} ,  ▷ \overline{α}^{1}, \overline{α}^{2} ∈ \overline{E},
77 20 RB ◁ 0 ≤ ϕ_{n} ≤ 1  ▷ ϕ_{n} ≥ 0
77 21 RB ◁ …where.  ▷ …where (see II.7.5).
77 -15 MR ◁ Means  ▷ Means.
77 -8 MR ◁ A = [ϕ > r],  ▷ A = [ϕ > r], ϕ ∈ E,
77 -6 RB ◁ Delete: “of functions”
77 -5 RB ◁ ψ_{n} = A · ϕ_{n}  ▷ 0 ∨ A·ϕ_{n} ∧ 1
77 -2 RB ◁ non–negligible  ▷ ∥ ∥^{∗}–non–negligible
78 1 RB ◁ integrable  ▷ integrable (for both means)
78 7 RB ◁ by f.  ▷ by f, provided the mean is σ–finite!
78 1 RB ◁ non–negligible  ▷ ∥ ∥^{∗}–non–negligible
78 8 MR ◁ integrable set  ▷ integrable sets
175 11 RB ◁ Replace the answer by the hint to apply 3.5 (iii)
175 12 RB ◁ C_{i}  ▷ C_{k}
78 14 RB ◁ Complex Valued  ▷ Complex–Valued
78 -6 MR ◁ almost every convergent ▷ almost everywhere convergent same on p79 l-6
79 6 RB ◁ \overline{E} ⊗ ℂ  ▷ \overline{E} ⊗ ℂ = \{ϕ + iψ : ϕ, ψ ∈ \overline{E}\}
79 8 RB ◁ und  ▷ and
79 -14 RB ◁ Banach Space Valued ▷ Banach Space–Valued
79 -11 RB ◁ add: , ∥f ∥_{E} : x ↦ ∥f(x)∥_{E}
80 8 RB ◁ complex valued  ▷ complex–valued
80 9 RB ◁ und  ▷ and
175 -20 RB ◁ After “there.” insert: Namely, ∥∫ ϕ∥_{E} ≤ ∑ ∥ξ ∥_{E} ∫ [ϕ = ξ] = ∫ ∥ϕ∥_{E} = ∥ϕ∥^{∗}.
80 -4 RB ◁ Extend: [Hint: use exercise II.5.16 on page 57.]
175 -15 RB ◁ E  ▷ \overline{E}
175 -14 RB ◁ measurable.  ▷ measurable (use 5.16).
80 -18 RB ◁ ∥f∥^{∗}  ▷ ∥ϕ∥^{∗}
81 -14 RB ◁ by the  ▷ by the permanence properties
82 13 RB ◁ lattice  ▷ vector lattice
83 4 MR ◁ S ∈ A  ▷ S ∈ A[ℝ]
83 -5 MR ◁ [f − f_{n} > δ]  ▷ [|f − f_{n}| > δ]
84 -13 RB ◁ delete “, say x ∈ A_{k},”
84 -11 RB ◁ for i  ▷ for all i
85 8 RB ◁ insert: ((M) reads here ρ_{0}(λ ∑_{k} ϕ_{k}) ⟶_{λ→0} 0 ⟹ ρ_{0}(ϕ_{k}) ⟶_{k→∞} 0 for ϕ_{k} ∈ E_{+} .)

86 -7 RB ◁ .)  ▷ . (delete spurious right parenthesis)

Chapter III Continued

Pa- Li- Au- ◁ Old  ▷ New  Com
ge ne thor◁ Text  ▷ Text  ment

86 -0 RB ◁ (Also MR)At the very end of the page insert this paragraph:

So far we have not specified in which sense the pointwise limits are to be taken, as limits of real–valued or of numerical functions, i.e. in ℝ^{S} or in \overline{ℝ}^{S}, S being the ambient space. Both make sense, and we shall correspondingly talk about the real sequential closure or the numerical sequential closure where the distinction needs to be stressed. We are generally interested in numerical functions, so E^{Σ} denotes the sequential closure of E in \overline{ℝ}^{S}, the numerical sequential closure. Unless the meaning is clear from the context we shall write E_{ℝ}^{Σ} for the real sequential closure, taken in ℝ^{S}, and E^{Σ} = EΣ ; \overline{ℝ} if it must be emphasized that the numerical E–Baire functions are meant. To make the distiction we also simply talk about real–valued or numerical Baire functions; this is justified by the following simple observation:

Exercise 5.0* If E consists of real-valued functions then E_{ℝ}^{Σ} contains the real–valued functions of E^{Σ} . (Answer: The real–valued functions in E^{Σ} form a real–sequentially closed family.)

These conventions require that the notation E^{Σ} be changed to E_{ℝ}^{Σ} where the real sequen-
tial closure is intended, e.g. in Proposition 5.3, Corollary 5.6 (i), etc.

87 8 RB ◁ E^{Σ}  ▷ E_{ℝ}^{Σ}  Same in lines 15 — 25

87 15 RB ◁ anyone  ▷ any one
87 -19 MR ◁ to ϕ ∗ f.  ▷ to ϕ ∗ f for all ϕ ∈ E.
88 15 KB ◁ chopping.  ▷ chopping of bounded functions

88 16 RB ◁ E^{Σ}  ▷ E_{ℝ}^{Σ}  Also in lines -14 — -11
88 -20 RB ◁ Delete: strictly  Also on line -18

88 -7 RB ◁ from 5.4  ▷ from 5.4 and 5.3
89 5 KB ◁ quite an  ▷ a real

89 9 RB ◁ Replace this and the next line with:

(ii): Let f be a function integrable for the mean ∥ ∥^{∗} + ∥ ∥^{♯}. If (ϕ_{n}) converges to f in this mean then it converges both in ∥ ∥^{∗}–mean and in ∥ ∥^{♯}–mean to f. Thus …

89 15 RB ◁ Theorem II.5.5  ▷ Theorems II.5.5 and II.7.5
90 2 MR ◁ (i) [f > r]  ▷ (i) [f < r]
90 3 MR ◁ (v) [f > d]  ▷ (i) [f < d]
90 7 MR ◁ M(F)  ▷ the family M(F)
90 8 RB ◁ ⋂_{N} ⋃_{n≥N}[f_{n} > r]  ▷ ⋃_{N,k} ⋂_{n>k}[f_{n} > r + 1/n]
90 16 MR ◁ Σ–algebras are  ▷ a σ–algebra is
90 -14 MR ◁ Delete: separable
91 4 MR ◁ Delete: Since for a … thus
91 15 MR ◁ One the  ▷ On the
92 8 RB ◁ negigible  ▷ negligible
92 16 RB ◁ delete E^{Σ}

92 -2 RB ◁ Delete the whole sentence up to (Exercise II.6.9). See p61, 4

93 16 RB ◁ < r.  ▷ ≤ r.
93 20 RB ◁ Proposition … (ii)  ▷ Proposition 5.7 (ii) on page 88
93 -6 RB ◁ Clearly  ▷ Since ∥ ∥^{↑} ∈ 𝔐

Chapter III Continued

Pa- Li- Au- ◁ Old  ▷ New  Com
ge ne thor◁ Text  ▷ Text  ment
94 7 MR ◁ E–Baire functions ▷ numerical E–Baire functions same in whole proof
94 16 MR ◁ Replace: (ii): According to (6.1) there are h_{n} ∈ E^{Σ} with |f| ≤ h_{n} and ∥h_{n}∥^{∗} ≤ ∥f∥^{∗} …
94 17 MR ◁ (Exercise II.7.7) ▷ as the limit of h_{n} ∧ k (Theorem 3.1 and MCT)
94 -16 RB ◁ Proposition … (i) ▷ Proposition 5.7 (i) on page 88
94 -11 MR ◁ the upper the  ▷ the upper
95 14 RB ◁ Carathédory ▷ Carathéodory same in headers
96 6 RB ◁ set function  ▷ set–function
97 7 RB ◁ analogue  ▷ analogous
97 11 RB ◁ (F_{n})  ▷ λ^{∗}(F_{n})
97 12 RB ◁ is increasing along ▷ is continuous along
97 17 RB ◁ Proposition II.3.2 ▷ Lemma II.3.1 and Proposition II.3.2
97 -8 RB ◁ restrictions…  ▷ restrictions to sets of known facts about functions.
98 2 RB ◁ with ∫^{∗}h < r  ▷ with h ≥ 1_{F} and ∫^{∗}h < r
99 -12 RB ◁ inequality  ▷ inequalities
100 3 RB ◁ II.2.17  ▷ II.7.1
100 22 RB ◁ page 99  ▷ page 98
176 -14 RB ◁ measure;  ▷ measure — such exists due to σ–finiteness
106 -6 RB ◁ This is  ▷ By Corollary II.5.6 on page 88 this is

Chapter IV

Pa- Li- Au- ◁ Old  ▷ New  Com
ge ne thor◁ Text  ▷ Text  ment
101 -5 RB ◁ Replace “that … do so.” by

that E is σ–finite and ∥ ∥^{∗} is continuous along arbitrary increasing sequences, and we shall do so.

176 -20 RB ◁ delete: ≤ 0

176 -5 KB ◁ Delete: II.6.9  see p61, l4

171 14 KB ◁ P and thus … ▷ P, and thus … insert a comma

Chapter V

Pa- Li- Au- ◁ Old  ▷ New  Com
ge ne thor◁ Text  ▷ Text  ment
103 -10 SB ◁ Delete: Minkowski’s inequality
103 -5 RB ◁ < b  ▷ ≤ b
105 9 RB ◁ is maximal  ▷ is maximal if ∥ ∥^{∗} is
105 -6 RB ◁ it  ▷ its conclusions
106 -7 RB ◁ ϕ_{n}  ▷ ϕ_{k}  twice
1646 9 SB ◁ UsingHölder’s…  ▷ Using Hölder’s…  insert a space

Indices

Pa- Li- Au- ◁ Old  ▷ New  Com
ge ne thor◁ Text  ▷ Text  ment
181 16 RB ◁ Insert: E_{+} the positive elements of E 3
182 1 Insert: F^{♮} the functions f with ∫^{♮}|f| < ∞ 17
182 -7 RB ◁ insert: R(E) the σ–ring generated by E 88
187 19 RB ◁ insert the line: Fatou’s lemma, 56
188 -4 RB ◁ measurable 72 ▷ measurable 72, 92
189 -3 RB ◁ MCT, 106  ▷ MCT, 54, 106
190 3 RB ◁ mean, 48, 90  ▷ mean, 48, 80, 90
191 13 RB ◁ Radon measure 140 ▷ Radon measure 80, 140

December 7, 2001