Suppose that Σ is a field of
subsets of the set S, and suppose that μ and γ are complexvalued finitely additive
set functions defined on Σ. Assume that μ is bounded and γ is finite and
absolutely continuous with respect to μ. (A word of warning is in order here. The
statement γ is absolutely continuous with respect to μ” is often interpreted as
“μ(E) = 0 implies γ(E) = 0”. This is not the meaning used here. Our definition is
“for every 𝜖 > 0 there is a δ > 0 such that _{β}(E) < δ implies γ(E) < 𝜖.”
Unless μ is bounded and countably additive, the two definitions are not
equivalent.)
THEOREM 1. There exists a sequence {f_{n}} of μsimple functions on S, such
that
 (1) 
uniformly for E ∈ Σ
 (2) 
where v(μ) is the total variation of μ.
Theorem 1 is established by a pure existence proof, and gives no indication of how
to find f_{n}. A more constructive result is given below.
A partition of S is a finite collection of sets E_{i} belonging to Σ, such that S is the
disjoint union of the E_{i}, and such that μ(E_{i})≠0,i = 1,⋯,n.
The set 𝒫 of partitions may be directed by refinement, that is, by the following
partial order: P_{1} < P_{2} if for every E ∈ P_{1} there exist F_{1},⋯,F_{r} ∈ P_{2} ( r may depend
on E) such that E and ⋃
_{i=1}^{r}F_{i} differ by a μnull set.
If P is a partition of S, define the μsimple function f_{p}^{γ} to be ∑
_{E∈P}(γ(E)∕μ(E))χ_{E},
where χ_{E} is the characteristic function of E.
THEOREM 2. If ∕ℓ is positive, then
uniformly for E ∈ Σ, where 𝒫 is directed as explained above.
