Correction to “Measurable gambling houses”
Ralph E. Strauch · Transactions of the American Mathematical Society · 1968
I am indebted to Mr. William D. Sudderth for pointing out an error in my paper Measurable gambling houses, Trans. Amer. Math. Soc. 126 (1967), 64-72. The definition of a measurable gambling house (Definition 1, p. 65) is not sufficient to insure the existence of a selection map a: F -* P such that a(f) E F(f) for all f The existence of such maps was implicitly assumed in the proofs of Lemma 1, Theorem 3, and the second result in Theorem 1, and those results are therefore incorrect as stated. The existence of such selection maps is a necessary and sufficient condition for the existence of measurable strategies and of random strategies (Blackwell and Ryll-Nardezewski, Non-existence of everywhere proper conditional distributions, Ann. Math. Statist. 34 (1963), 223-225). A gambling house F is said to be leavable if 8(f) E F(f) for all f, where S(f) is the gamble which assigns probability one to the fortune f. If F is leavable, the required selection maps exist, and the above theorems are true. The second conclusion of Theorem 1 (p. 66) should therefore be modified to read