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

328 MARCUS AND RIIZEI.
Proof. The odd and even parts of X each satisfy the assumptions of the
lemma. Therefore it is sufficient to prove the lemma in the case where .iy’ is
either odd or even.
Suppose that Z is odd. Let a bc a real number and let C; be a set in 7 such
that there exists a simple function g with K(g) n 1: 8:. such that
where p is a Lyapunov measure associated with ~ti. Then we define
H,(., a) =- Af(UX~ ~- &-o! . (8.6)
In view of Lemma 8.2 the right-hand side of (8.6) does not depend on the
choice of the function g. Thus H,( ., u) is well defined. If F is a measurable
subset of l!, then H,( ., a) is also defined; the function g -~ a~~.,,,. has the
properties required in order that H,(., a) be defined. Moreover, we have
Thus,
H,(., 4X,J = *qaxLT - R)XV = S(axr. - (g - ~Xu\v))Xr~ .
H,(., u) = I{,(., a)xv. (X.7)
As a consequence of (8.7), if { I’, , L’,} is a partition of U, we have
II,(., u) =- Hul(*, a) -+ El&(., a). VW
Now let U be an arbitrary set in 7 and let (C; , U,} and {I’, , I~VT) be partitions
of U into sets of equal p-measure. ‘Then HVt(., u) and Hvi(*, a), i 1. 2, arc
defined. We claim that
This is proved as follows. Let Wj,; mm= Uj n lTj (i, j -- 1, 2). Let i lI’P’,j ( II’: ;j
be a partition of W,,j into sets of equal p-measure. Set
Note that U,’ u U,’ = Vi’ v ET2’ and denote this set by C:‘. Similarly
U: v UG = I,‘: v I/‘; and we denote this set by 0’“. We also have p( I -‘) jl( U”)
so that H,,(., a) and H,-(., u) are defined. By (8.8),