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220 XIII INTEGRATION

(1) Uniqueness. Every compact subset of X x Y is contained in one
of the form L x M, where L and M are relatively compact open subsets of X
and Y respectively ((3.20.17) and (3.18.2)). We shall show that, for each
function h e 3tf_c(X x Y) with support contained in L x M, the value of
v(h) is well-determined. This will follow from

(1 3.21 .1 .2) If L c X and M c Y are relatively compact open sets, then every
function h e JT_C(X x Y) with support contained in L x M is a cluster point,
in the Banach space Jf*_c(X x Y; L x M), of the set of functions of the form

(x_yy)^>^fi(x)g_k(y\ where (/,) is a finite sequence o unctions in J_cX

i,k

with support contained in L, and(g_k) is a finite sequence of functions in «?f _C(Y)
with support contained in M.

Assuming this for the moment, by hypothesis there exists a number c> 0
such that | v(w)| ^ c * \\u\\ for all u e Jf_c(X x Y) with support contained in
L x M. On the other hand, for each s > 0 there exist two finite sequences of
functions /_£ e tf_c(X)₉g_k e Jf_c(Y) such that

for all (x, y) e L x M, the support of the left-hand side being contained in
L x M. Using (13.21.1.1), it follows that

i, k

and since s is arbitrary, this establishes the uniqueness of v.

To prove (13.21.1.2), let e be a positive real number. Then for each z =
(X y) e L x M there exists a compact neighborhood U c L (resp. V c M)
of x in X (resp. of y in Y) such that the oscillation of h in U x V is :ge
(3.20.1). The projections S c L and T c= M of Supp(/z) are compact, hence
for each x e S there exists a finite number of points yfa) (1 ^y ^ n(x)) of T,
and for eachy a compact neighborhood U/x) cz L of x in X and a compact
neighborhood V/.y/X)) c M of y^(x) in"Y, such that the oscillation of h in
Vj(x) x Vj(yj(x)) is g e, and such that the interiors of the sets Vj(yj(x))
cover T. The set W(x) = Q U/(x) is a compact neighborhood of x in X.

Hence there exists a finite number of points x_t (1 ^ i g m) in S such that the
interiors of the sets U'(.Xf) cover S. Put A_f = U'(*i), and let (B_k)i^k^_P be the
family of sets obtained as follows: for each point y E T, let W(y) be the inter-
section of the (finitely many) interiors of sets VjO^jCj)) which contain y; the
sets WOO ^are °P^en ^and nonempty, and form a finite open covering of T which
we denote by (B_fc)^_fc^_p. Notice also that, by construction, the oscillation of
h in each set A_{ x B_fe is ^ e. Now let (/^ ^ _f^_w (resp. (g^)_l ^_k^_p) be continuous
mappings of X (resp. Y) into [0, 1], such that Supp(/_f) c A₄ and Supp(^) c B_fc



	220 XIII INTEGRATION (1) Uniqueness. Every compact subset of X x Y is contained in one of the form L x M, where L and M are relatively compact open subsets of X and Y respectively ((3.20.17) and (3.18.2)). We shall show that, for each function h e 3tf_c(X x Y) with support contained in L x M, the value of v(h) is well-determined. This will follow from (1 3.21 .1 .2) If L c X and M c Y are relatively compact open sets, then every function h e JT_C(X x Y) with support contained in L x M is a cluster point, in the Banach space Jf_c(X x Y; L x M), of the set of functions of the form*

	(x_yy)^>^fi(x)g_k(y\ where (/,) is a finite sequence o unctions in J_cX i,k with support contained in L, and(g_k) is a finite sequence of functions in «?f _C(Y) with support contained in M. Assuming this for the moment, by hypothesis there exists a number c> 0 such that \| v(w)\| ^ c * \\u\\ for all u e Jf_c(X x Y) with support contained in L x M. On the other hand, for each s > 0 there exist two finite sequences of functions /_£ e tf_c(X)₉g_k e Jf_c(Y) such that

	for all (x, y) e L x M, the support of the left-hand side being contained in L x M. Using (13.21.1.1), it follows that

	i, k and since s is arbitrary, this establishes the uniqueness of v. To prove (13.21.1.2), let e be a positive real number. Then for each z = (X y) e L x M there exists a compact neighborhood U c L (resp. V c M) of x in X (resp. of y in Y) such that the oscillation of h in U x V is :ge (3.20.1). The projections S c L and T c= M of Supp(/z) are compact, hence for each x e S there exists a finite number of points yfa) (1 ^y ^ n(x)) of T, and for eachy a compact neighborhood U/x) cz L of x in X and a compact neighborhood V/.y/X)) c M of y^(x) in"Y, such that the oscillation of h in Vj(x) x Vj(yj(x)) is g e, and such that the interiors of the sets Vj(yj(x)) cover T. The set W(x) = Q U/(x) is a compact neighborhood of x in X. Hence there exists a finite number of points x_t (1 ^ i g m) in S such that the interiors of the sets U'(.Xf) cover S. Put A_f = U'(i), and let (B_k)i^k^_P* be the family of sets obtained as follows: for each point y E T, let W(y) be the inter- section of the (finitely many) interiors of sets VjO^jCj)) which contain y; the sets WOO ^are °P^en ^and nonempty, and form a finite open covering of T which we denote by (B_fc)^_fc^_p. Notice also that, by construction, the oscillation of h in each set A_{ x B_fe is ^ e. Now let (/^ ^ _f^_w (resp. (g^)_l ^_k^_p) be continuous mappings of X (resp. Y) into [0, 1], such that Supp(/_f) c A₄ and Supp(^) c B_fc