A note on a theorem by H. D. Brunk

Abstract

Throughout the paper, the notation will be consistent with that used by H. D. Brunk in [1]. That is (Q, 8, ,u) is a complete measure space, and L2 denotes the set of square integrable functions corresponding to it. We shall call 2, a collection of sets in 8, a sub-cr-lattice if it is closed under countable unions and intersections, and contains the null set 0, and Q. A function X is ?-measurable if [X>a]E,C for all real a. L2(SC) denotes the set of ?-measurable functions which are also in L2. A family C of measurable functions is called a convex cone if k _ 0, X E C, YE Cz?>kX E C, X + YE C. A collection of functions is a lattice if the pointwise supremum and infinum of any two functions in the collection is in the collection. If M is a collection of functions, -M= {-X: X&CM}. Similarly, Cc= {A: AcE42} . IA or I(A) will be the indicator function of the set A. In [1], H. D. Brunk stated the following theorem.