Lebesgue theory on a Boolean algebra
John M. H. Olmsted · Transactions of the American Mathematical Society · 1942
A necessary and sufficient set of conditions that a function, E(a), from real numbers to subsets of a set S, be associated, as above, with a real-valued point function everywhere finite is 1. E(a) 4, as a T 21 En(l -n) = S,1 O' 1E (n) =0O; 3. EZ l2 E(a +1/n) =E((a) for every a. It is possible, therefore, to consider f(x) and E(a) as different aspects of the same function, one more natural for algebraic combinations of functions, and the other receiving emphasis in a theory of integration. However, one may restrict oneself entirely to the second aspect. For example, if A(a) =E [f(x)> a],