Gruber P. Convex and Discrete Geometry

Lyapunov’s Convexity Theorem
4 Support and Separation 61
Let M be a set with a σ -algebra M of subsets. A finite signed measure ν on the
measure space 〈M, M〉 is non-atomic if, for every set A ∈ M with ν(A) �= 0, there
is a set B ∈ M with B ⊆ A and ν(B) �= 0,ν(A). Lyapunov [671] proved the
following basic result.
Theorem 4.5. Let µ1,...,µd be d finite, signed, non-atomic measures on 〈M, M〉.
Then the range of the (finite, signed, non-atomic) vector-valued measure µ =
(µ1,...,µd) on 〈M, M〉, that is the set
is compact and convex.
R(M) = � µ(B) = � µ1(B),...,µd(B) � : B ∈ M � ⊆ E d ,
There exist several proofs of this result. We mention the short ingenious proof of
Lindenstrauss [659]. Below we follow the more transparent proof of Artstein [40].
Let relbd stand for boundary of a set relative to its affine hull.
Proof. Before beginning with the proof, some useful notions and tools will be presented:
The variation |µ| of the vector-valued measure µ is a set function on 〈M, M〉
defined by
� �n
|µ|(A) = sup �µ(Ai)� :A1,...,An ⊆ A, disjoint,
i=1
�
A1,...,An ∈ M, n = 1, 2,... .
|µ| is a finite, non-atomic measure on 〈M, M〉.Thus,anysetA∈Mwith |µ|(A)>0
contains sets B ∈ M for which |µ|(B) is positive and arbitrarily small. A finite
signed measure ν on 〈M, M〉 is absolutely continuous with respect to the measure
|µ| if ν(B) = 0 for each B ∈ M with |µ|(B) = 0. This is equivalent to each of the
following statements: first, for each ε>0, there is a δ>0 such that |ν(B)| ≤ε for
any B ∈ M with |µ|(B) ≤ δ. Second, there exists a function f : M → R which is
integrable with respect to the measure |µ| such that
�
ν(B) = fd|µ| for B ∈ M.
B
f is the Radon–Nikodym derivative of ν with respect to |µ|. GivenA ∈ M, let
A − , A 0 , A + denote the following sets:
A − ={s ∈ A : f (s) 0},
all in M. Clearly, |µ|(B) = 0 for each set B ∈ M, B ⊆ A + , with ν(B) = 0. That
is, the restriction of |µ| to A + is absolutely continuous with respect to the restriction
of ν to A + and similarly for A − .