Extension Theorems of Hahn-Banach Type for Nonlinear Disjointly ...

308 MARCUS AND MIZEI.
PROPOSITION 3.4. Let ~2’ be a rich subspace of L”(m). Then. there e.rists a
vector measure p: 7 + I1 such that
I. TV has bounded total variation (which we denote by j p I);
II. TV is absolutely continuous with respect to m;
III. p is a I.yapunov measure;
IIT. &? =: {f ELm(m) : Snf dW = O}.
Conversely, if TV: 7 ---+ l1 is a Lyapunov measure satisfying conditions I and II
and if AS? is de$ned by IV, then Jd is a rich subspace.
A measure p satisfying the conditions of Proposition 3.4 will bc called a
Lyapunov measure associated with JY.
Given a Lyapunov measure, one can construct related measures which arc
also of Lyapunov type. The f o 11 owing proposition describes two such
constructions.
PROPOSITION 3.5. Let TV: 7 - l X be a Lyapunov measure which is absolutely
continuous with respect to m.
(a) If gl ,..., g,. are in Ll(m) and p: 7 + Rk :< S is the measure given bzl
then j.2 is a Lyapunov measure.
F(E) .-z (lEgl dm ,... , .kg, dnz, p(E)), VE E 7, (3.1)
(b) If u is a p-integrable function and af 1/U is the indefinite integral of u z&h
respect to p, then vl, is a Lyapunov measure.
For the proof of part (a) see [9; Sect. 21 and for the proof of part (1~) see
[4; Chap. V, Sect. 21.
We now prove the following statement which was mentioned in Section 2.
LEMMA 3.6. (a) If J&’ is a closed linear subspace of LY(m), 1 ; p < z,
such that A? is of $nite codimension, then At is a rich subspace.
(b) If .Ad is a w*-closed linear subspace of Lx(m) of Ji ni t e codimension then A
is a rich subspace.
Proof. Under the above assumptions the annihilator of JZZ’ in L”(M), (1 /p) +
(l/q) :: I, is finite dimensional. i2s before, denote the annihilator by k’ .
Clearly, by assumption (a) or (b), the annihilator of .~&‘l in Ln(m) is precisely JZ.
Let (ql ,..., pk) be a basis for J&‘~ and set