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

312 MAKCUS AND MIZEL
The proof of the theorem for odd S relies on the following Radon Sikodyn
type result which was established in [9].
PROPOSITION 4.3. Let ,//I’ be a rich subspace of L=(m) and let ‘1 be an odd
functional on ~?j[ uhich is disjointly additive and (bm)-continuous. Then there
exists a function G in Ll(m) such that
Let A! be a rich subspace of L%(m) and let p be a Lyapunov measure associated
with A?. A functional X on my,/ is said to be t”-invariant if, for ever!. pair of
functions f, g in .y,[ such that
we have N(f) == X(g).
t4.f ‘(4) P.(‘TW), VaER, (4. IO)
Using Proposition 4.3 we shall reduce the proof of the theorem for odd ;V to
the case where N is, in addition, p-invariant.
LEMMA 4.4. Let ,k/ and JV be as in Theorem 4.1 and suppose that S is odd.
Then there exists a function G: Q K R --f R such that
and
G(., a) EL’(m), tin E R,
G(., a) r= -G(*, -a), Va E R,
(4.11)
,V(ah) =~. 1. G(., a) 17 dm, Qh E &,, and VaER. (4.12)
I “0
Furthermore, if p is a LJ,apunov measure associated with & and fi is a Sfunrtional
on 9’;( defked by,
then fi is p-invariant.
Proof. Given a real number a, set
A,,(h) = h:(ah), Q’h E cl’,, . (4.14)
Then A! and A,, satisfy the conditions of Proposition 4.3. Hence, there exists
a function G(., a) in Ll(m) such that (4.12) holds. Since A,, mu I 1 ,, . one can
select the function G in such a way that (4.11) holds.
It remains to be shown that the functional 2%’ defined by (4.13) is ~-invariant.