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

334 MARCUS AND iVIIZEL
Sow let ~2’ CLV-(m) be a subspace possessing the properties
(a) ,&’ contains a function uu bounded away from zero,
(b) whenever f, g E JZZ are nonnull, then .f. g is nonnull.
Let G be an arbitrary (bm)-continuous functional on ~2’. Put
where
Xf,(u) = aG(w), (9.8)
20 = u - av() ) a = (1’ uv,, dm) \,(J v()2 dm).
Clearly, NO is (bm)-continuous on JY and, vacuously, disjointly additive (here
we use (b)).
Finally, consider the following functional F,, defined on the subspace /2” C i ti
consisting of functions za such that j ZU’Z’~ dnz mm: 0.
F,,(w) = (.n2iVo(av, + w) da = (a, - a,) G(w),
. “1
w E J&i?‘. (9.10)
Supposing that ATO has a (hm)-continuous, disjointly additive extension to all
of L”(m), it follows by Lemma 9.1 that F,, is G-differentiable everywhere in .J
with respect to functions in ~2’. However, the functional G was arbitrary so that,
in general, this differentiability property fails to hold. Hence, in general, ATO will
not possess an extension of the type sought.
As a consequence of the above, a class of subspaces lacking the d.a. extension
property can be constructed, for instance, by taking the uniform closure of any
linear manifold consisting of harmonic functions (and including constants) on a
domain Q C R”. Here m is n-dimensional Lebesgue measure.
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