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

314 MARCUS AND MIZEL
Note that condition (c) is weaker than (bm)-continuity. We defer the proof
of this lemma to the next section and proceed now with the completion of the
proof of Theorem 4.1. We need one additional result which was obtained in
[9; Theorem 3.11.
PROPOSITION 4.6. Let & be a rich subspace of Ll(m). T’hez sp C,,( the
linear span of rZfl) is dense in A![ with respect to the Lr norm.
The existence of a kernel for N follows by the following argument. If ;V is
odd and fi is defined by (4.13) then l%r satisfies conditions (a))(c) of Lemma 4.5.
Therefore the function H : R + R, defined by H G 2 I^r with G as in
Lemma 4.4 and l? as in Lemma 4.5, is a kernel for N. If N is even, the existence
of a kernel is given by Lemma 4.2. As previously mentioned, the existence of a
kernel in the cases where X is either odd or even, implies its existence in the
general case.
In order to prove the uniqueness statement in Theorem 4.1 we have to show
that any function H: Sz x R + R satisfying (4.1) such that
i’ H(f) dm 0, Vf E ?I > (4.15)
R
is, essentially, of the form H, for someg E A-. For the definition of H, . ~~(2.2).
Let H be a function as above and set
H,,(., a> = [H(., a) - H(., -a)]i2,
H,(., a) ~~ [H(., a) -I- H(., -a)]/2.
(4.16)
Let p be a Lyapunov measure associated with AC. Let U E 7 and let {Ur , U,jbe
a partition of U into sets of equal p-measure. Then,
s
by (4.15) and (4.16),
H,(., a) dm : Jb, Hk, 4 dm -t ju. Hk, -a)
u 2
elm
Thus,
1
z = 2 [I, fJ(., 4 dm + Ju,, HC.9 -4 dm]
1
+ i [jul H(*, -a) dm -i- ju2 H(., a) dm] == 0.
Zi,(., a) Mom 0, Va E R.
Next, (4.15) and (4.16) imply, in particular, that
jD H,(., a) h dm 0 Vh E CT,, , Vu E R.
(4.17)
(4.18)