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

324 MARCUS AND MIZEI.
LEMMA 7.3. Let {f?,) be a sequence of simple functions such that .f,, - + 0 i/z
I,“(m). Then,
!jz lo H(fn) dm 0. (7.2)
Proof. Our assumption on {fn) implies that safn cI~ --+ 0. Hence, by Lemma
7.2, there exists an integer n, and a sequence of functions {h,f,,;),,, , each of
which is a difference of characteristic sets, such that
.I’ fil dp
n
[ h, dp for 72 .‘; 12” and m(K(h,)) --f 0. (7.3)
- R
By Lemma 3.7 we can construct p-uniform decompositions of fiL and h,, , say
{fn’, f3 and {h,‘, hi’J such that K(fn’) n K(h,“) K(f,!J n K(h,,‘) G. Set
g, 7 fn’ - h,‘, n 2: n, . Then g, E q(, (n > n,,) and g, ---f 0 in L”(m). Ry
Theorem 4.1 and the continuity of AT, we obtain
Wg,) j3, W,) dm, n ;;- I+,
However, the properties of h,,’ imply that
Since f,,’ and 11,’ are disjoint we have,
Hence, by (7.4),
Similarly, one shows that
km= [ H(-h,‘) dm : = 0.
’ ‘R
H(g,) == H(fn’) -+- H(--A,‘).
i+i 1 H(fi) dm = 0.
u
and lim N(g,) :=I 0.
?1 .> rl
Since H(fJ = H(fn’) -t H(fi), th e p roof of the lemma is completed.
LEMMA 7.4. Let {.frL} be a sequence of simple functions which converges in
L”(m) to a function f in A’. Then,
Proof. The assumption on (fn} implies that,
(7.4)
l,‘fl .i, H(fn) dm -= iv(f). (7.5)