A representation problem
David Blackwell · Proceedings of the American Mathematical Society · 1954
since f(x) _ ? c = (1/a)fJf(x)dx. The result of this paper is that (2) is sufficient as well as necessary for a function f to admit a representation (1). The problem arises in connection with a submarine search game discussed by Morse and Kimball [1], which runs as follows. A submarine must pass through a channel of length 1, and must choose a part of the channel of length a at which to be on the surface. An enemy plane must choose a point x along the channel at which to look for the submarine. The submarine escapes if it is submerged at the x chosen by the plane; otherwise it has a probability h(x) of being detected. Thus a pure strategy for the submarine is a subset S of 0 <x ? 1 of Lebesgue measure a, and its detection probability for the given S as a function of x is h(x)4(x), where 4 is the characteristic