Comments for  “Measurable Weak Sections”  G. A. Edgar

Correction, page 639, line 6. Replace “Thus” to the end of the proof with the following.

Thus

u^{−1}(G) ⊆ { s ∈ S : s ∈ Φ′(G) } = Φ′(G).

But an open Baire set is a countable union of closed Baire sets, and Φ is a σ-homomorphism, so u^{−1}(G) ≡ Φ′(G) [μ] for open Baire sets G. But ℬ (X) is generated by the open Baire sets, so u^{−1}(A) ∈ Φ(A) for all A ∈ ℬ (X).

Prop. 1.4 “Lifting of measures”. See:

N. Bourbaki, Intégration, Ch. IX, §2, no. 4, Prop. 8. (1969)

Lembcke, Konservative Abbildungen und Fortsetzung regulärer Maße.

Bauer, Konservative Abbildungen lokal-kompakter Räume. Math. Ann. 138 (1959) 398–427.

Theorem 2.3. Parts (b) and (c) are doubtful.

Prop. 3.4 Probably Borel measurability holds. See Babiker & Knowles, Mathematika 25 (1978) 120–124.

Theorem 3.8. Alternate proof by M. Talagrand.

related papers.

J. Lehn (uniqueness of preimage measures)

H. v. Weizsäcker

S. Graf, Measurable Weak Selections

S. Graf, A parametrization of measurable sections via extremal preimage measures D. H. Wagner, Survey of measurable selection theorems. Siam J. Control & Opt 15

(1977) 859–903.

A. D. Ioffe, Survey of measurable selection theorems: Russian literature supplements. Siam J. Control & Opt 16 (1978) 728–732.

Leese, Measurable selections and the uniformization of Suslin sets. Amer. J. Math.

Rieder, Measurable selection theorems for optimization problems. Manuscripta Math. 24 (1978) 115–131.

W. Hackenbroch, Dilated selection theorems and extremal preimage measures. Arch. Math. 43 (1984) 434-439.