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

330 MARCUS AND MIZEL
Here we used (8.1 I), (8.6), and the fact that SF is local and disjointly additive.
Similarly, we obtain
WtJ Wfd (8.13)
By the disjoint additivity of H and .X, (8.12) and (8.13) imply (8.5).
Next we consider the case where 2 is even. Let U be a set in 7 and let [I .i, L.,],
(E,‘, CT,‘> be two partitions of li into sets of equal p-measure. 1Ve claim that
By the disjoint additivity of X and (8.15),
Jwx u, - XUJ) = Je(Xw,,, - xw,,J -- m4Xw,,, - XWJL
=@Mx U,’ - X$‘)) =- x(u(x~l,, - xw,,2)) -t- ~~MXW,,, -- xw,,,)).
In view of the fact that X is even these equalities imply (8.14). Let i. and
{Cj’r , Uz} be as above, and set
ffu(-, a) = JWXU, - XUJ), Vu E R. (8.16)
By (8.14) the right-hand side of (8.16) d oes not depend on the choice of the
partition. Let H(., u) HsL(., u) for every u. Then
I{,(., 0) =- ff(., a)xc , Vl-~r, VaER. (8.17)
Indeed,
additivity,
if CC”, 1,. “‘] is an arbitrary partition of I - then, by (8.16) and disjoint
]I,.(., u) FI,.,(., 0) L- H,.-(., u), Vu E R.
In particular, we have
I-i(., 0) = HA., a) + ffc,,,-(., n), Vu E R.
Since X is local, (8.16) implies that K(1l,-(., 0)) 1; L.. ‘l’his relation together
with the above equalitv viclds (8.17).
. I