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

Page 664 Line 17. Replace cl_{w} by cl_{w} Page 664 Line 18. Replace cl_{w} by cl_{w} Page 665 Line 4. Replace eaxmple by example Page 668 Line 2. Replace [22] by [2] Page 668 Line 5. Replace [22, Lemma 9] by [2, Lemma 9] Page 668 Line 6. Replace 2 1 by 2.1 Page 668 Line 20. Replace Baire by Baire Page 670 Line -14. Replace τ-smooth by τ-smooth Page 672 Line 14. Replace countable by uncountable Page 673 Line 2,3. Replace weak by weak (three times) Page 673 Line 6. Replace ba by be Page 677 Line 11. Replace in by In: Other remarks

1.3: YES, Talagrand & Fremlin.

1.4: In fact, the identity is universally Luzin measurable in both directions.

1.5: Same.

1.5: See also: Lemma 2.6 of J. Hoffmann-Jorgensen, The strong law of large numbers and the central limit theorem in Banach spaces. [Proceedings of the Seminar on Random Series, Convex Sets and Geometry of Banach Spaces (Mat. Inst., Aarhus Univ., Aarhus, 1974; dedicated to the memory of E. Asplund), pp. 74–99.]

1.4, 1.5: See Theorem 1 of V. I. Rybakov, Some properties of measures on a normed space that has the RN property. (Russian) Mat. Zametki 21 (1977), 81–92.

1.5, 1.7: See R. D. Bourgin’s book, 4.3.11.

1.6: Fremlin says Lemma 1.6 surely holds for K-analytic.

1.7: Also: If Borel(X^{∗}, norm) ⊆ Univ(X^{∗}, weak*).

1.7: See Saab.

2.1 and Bockstein: Theorem 4 of Ross & Stone, Products of measure spaces. Amer. Math. Monthly 71 (1964) 398–403.

2.1: See Engelking, On functions defined on cartesian products. Fund. Math. 59 (1966) 221–231. Also A. W. Hagen, Approximation of real continuous functions on Lindelof space. Proc. Amer. Math. Soc. 22 (1969). It follows that: If every countable sub-product is Lindelof, then Baire(∏ X_{i}) = ∏ Baire(X_{i}).

5.2: Tortrat’s result actually says a measure on Baire(X, weak) must have separable support. Hence we can avoid Grothendieck here. Grothendieck is a corollary of Tortrat. Similarly for 1.4 use Tortrat.

J. Uhl, Vector valued functions equivalent to measurable functions. Proc. Amer. Math. Soc. 68 (1978) 32–36.

Odell & Rosenthal, A double-dual characterization of separable Banach spaces containing l^{1}. Israel J. Math. 20 (1975), 375–384. l_{1} does not embed in X if and only if Baire(X^{∗}, weak*) = Baire(X^{∗}, weak).

Hayden: Even if X is nonseparable, l_{1} does not embed in X ⟺ Baire(X^{∗}, weak) ⊆ Univ(X^{∗}, weak*).

Typeset by AMS-TEX

Dubins & Freedman, Measurable sets of measures. Pacific J. Math. 14 (1964) 1211–1222. μ ↦ μ(A) measurable for all A; countably generated case.

A. Janssen, Messbare Mengen von Massen und Inhalten. Man. Math. 34 (1981) 1–15. Baire is the σ-algebra generated by the maps μ ↦ μ(A).

Rybakov (cited above), Theorem 4: l^{1} does not embed in X if and only if every vector measure in X′ has relatively compact range. Theorem 5: the Schwartz theorem plus mine (Theorems 1.5, 1.7).

Theorem 2.3 can be used to prove two results in Goldman & Talagrand, Math. Scand. 44 (1979) 345–356. See Remarque, p. 348.

Noble & Ulmer, Factoring functions on Cartesian products. Trans. Amer. Math. Soc. 163 (1972) 329–339. Depends on countably many coordinates ⟺ pseudo-ℵ_{1}-compact.

1.5 used in: R. Alencar, Multilinear mappings of nuclear and integral type. Proc. Amer. Math. Soc. 94 (1985) 33–38.

J. de Maria, The space l_{∞}/c_{0} is not a Radon space. Proc. Amer. Math. Soc. 112 (1991) 1095–1100.

A. Bozorgnia & M. Bhaskara Rao, Measurability and convergence theorems for random elements in Banach spaces in terms of total subsets of linear functionals. Bull. Inst. Math. Acad. Sinica 14 (1986) 383–416. Summarizes different kinds of measurability and convergence.