 
Summary: 1. Algebras of sets and the integration of elementary functions.
Let X be a set.
Definition 1.1. Suppose A is a family of subsets of X. We let
u(A) = {F : F A and F is finite};
i(A) = {F : F A and F is finite};
c(A) = {X A : A A}.
We also set
d(A) = {F : F A, F is a finite and F is disjointed}.
We say A is an algebra of subsets of X if the following three conditions hold:
(U) u(A) A; (I) i(A) A; (C) c(A) A.
Note the redundancy: U and C imply I and I and C imply U by the DeMorgan
Laws. Note also that the union of the empty family of subsets of X is the the empty
set so an algebra of subsets of X always contains and X.
Definition 1.2. Suppose S is a family of subsets of X. Note that 2X
is an algebra
of subsets of X and that S 2X
so that
{A : A is an algebra of subsets of X and S A} = .
Thus
a(S) = {A : A is an algebra of subsets of X and S A}
