Errata for “Measurability in a Banach space, II” G. A. Edgar

Page 563 Line 6. Replace that by than

Page 563 Line 11. Delete 3.1

Page 565 Line 3. Replace 6 by 5.6

Page 565. Replace Proposition 3.2 by

3.2’ Proposition. If X has the PIP and the Radon-Nikodým property, then X is measure-compact.

Proof. Let μ be a probability measure on Baire(X, weak). I must show that μ is tight. Let ε > 0. There is M < ∞ so that Ω = { x ∈ X : ∥x∥ ≤ M } has outer measure > 1 − ε. Define ℱ = { A ∩ Ω : A ∈ Baire(X, weak) }. [This is not, in general, equal to Baire(Ω, weak).] Define a finite measure ν on ℱ by ν(A) = μ^{∗}(A) for all A ∈ ℱ. Then the function φ: Ω → X defined by φ(x) = x is bounded and scalarly measurable. So by the PIP, the vector measure defined by

m(A) = ∫_{A} φ dν, A ∈ ℱ

exists. By the RNP, there is a Bochner integrable function ψ: Ω → X, weakly equivalent to φ, such that

m(A) = ∫_{A} ψ dν, A ∈ ℱ.

Therefore ν = ψ(ν) = φ(ν) is tight, so there is a compact set K with μ^{∗}(K) ≥ 1 − 2ε. This shows that μ is tight. □

Page 572. Replace (2) ⇒ (1) by

(2) ⇒ (1). A completely regular space is realcompact if and only if every zero-one Baire measure is tight. Assume (2) holds, and let μ be a zero-one measure on Baire(X, weak). There is a ball B with positive outer measure, and hence outer measure 1. Let Ω = B, ℱ = { A ∩ B : A ∈ Baire(X, weak) }, and define μ̃ on ℱ by μ̃(A ∩ B) = μ(A). Then the identity function φ: Ω → X is a bounded scalarly measurable function, so by assumption it is Pettis integrable. Let x = ∫ φ dμ̃. We claim that μ is the point-mass ε_{x}. The measures μ and ε_{x} agree on all sets of the form { ω : x^{∗}(ω) ≥ t } which generate Baire(X, weak) [6, Theorem 23], so μ = ε_{x} since both are zero-one measures. This shows that (X, weak) is realcompact. □

Page 573 Line 21. Move (See note added in proof.) to the end of the previous paragraph.

Page 577 Line -9. Replace e_{n} by e_{n}^{∗}

Additional Remarks

Theorem 5.9’ For any set Γ, the space l^{1}(Γ) has the Lebesgue-PIP.

Proof. The proof in the paper (Theorem 5.9) may be used if 𝔪_{r} ≥ 𝔠 [we only need 𝔪_{r} > 𝔠, which is still easier].

If 𝔪_{r} ≤ 𝔠, then Axiom K holds (Kunen’s thesis, Theorem 14.7): There is a set of reals of cardinality < 𝔪_{r} which is not Lebesgue measurable and the set of real numbers is not

Typeset by AMS-TEX

the union of fewer than 𝔪_{r} sets of Lebesgue measure 0. Then, by Fremlin & Talagrand [8, Proposition 3C], every Banach space has the Lebesgue-PIP. □

An interesting use of Pettis integral: A. B. Guliasašvili, Estimates for the Pettis integral in interpolation spaces, and a generalization of some imbedding theorems. Soviet Math. Dokl. 25 (1982) 428–432.

M. Talagrand, Pettis Integral and Measure Theory. Memoirs of the A. M. S. 307. Among many other results, we have:

16-2-6: l^{1}(Γ) is measure-compact if and only if card Γ < 𝔪_{r}. l^{1}(Γ) is realcompact if and only if card Γ < 𝔪_{2}.

16-4-1 (p. 199): An example X measure-compact, not (C), not Mazur.

G. Vera, Caracterizacion de la Madida Imagen de una Funcion Integrable Pettis, 1985. φ is Pettis integrable if and only if the induced measure λ satisfies:

(*) For every net (Z_{j}) of convex zero-sets that decreases to the emptyset, λ(Z_{j}) → 0. (In Spanish, so I did not read it.)

R. Pol, Note on the spaces P (S) of regular probability measures whose topology is determined by countable subsets. Pacific J. Math. 100 (1982) 185–201. The dual ball of D[0, 1] is the continuous image of a first-countable space. (p. 16 and footnote 10)