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284 XIV INTEGRATION IN LOCALLY COMPACT GROUPS
for all s E K (3.16.5). Hence, for all x e V, we have

{* r

I /(s-'*o) dn(s) g 2 H/ll M(JK) + |

from which (ii) follows.

(iii) Again we may suppose that fi ^ 0. By hypothesis, for each e > 0,
there exists a compact subset H of G such that |/(x)| ^ e for all x ^ H. Take
K as in the proof of (ii) above, and suppose that x $ KH. Then if s e K we
H and therefore

JG-K

^ ll/ll '/

which proves that \JL */e

(14.9.3) £t?ery measure jn on G zj convolvable with every function f e jf~_c(G);
r/ze integral on the right-hand side of (14.8.2) is defined for all x e G, awrf */ze
function x\-+j f(s~~¹x) d^(s) is continuous on G.

Since the measure/- ft has compact support, p and/* ft are convolvable
(14.6.4), and it is clear that the integral tf(s~~¹x) dfj,(s) is defined for all x G G.
The continuity of the function x*-*jf(s~^lx) dfi(s) follows from (14.1.5.5).

We shall leave to the reader the task of stating the corresponding proposi-
tions for the convolution/* p. It should be noticed in particular that (14.9.2)
and its analog for/* /z prove that if G is unimodular (14.3), then «S?g(G) is a
left and right module over the algebra M£(G), and the external laws of com-
position of these two module structures are compatible by virtue of (14.7.2).

(14.9.4) Let p, v be two measures on G, and let fe Jf_c(G). Suppose that
ji and v are convolvable. Then the function n*fis v-integrable and

(14.9.4.1) </,/**v> = <Ai*/,v>.

Likewise, if p and v are bounded and feV$(G), the function /u */ is
continuous and bounded (hence v-integrable) and the formula (14.9.4.1) is valid.

For the hypothesis implies that the function (s₉ x)\-*f(s~^lx) is (^ ® v>
mtegrable, and the result therefore follows from the theorem of Lebesgue-
Fubini.



	284 XIV INTEGRATION IN LOCALLY COMPACT GROUPS for all s E K (3.16.5). Hence, for all x e V, we have

	{* r I /(s-'o) dn(s)* g 2 H/ll M(JK) + \|

	from which (ii) follows. (iii) Again we may suppose that fi ^ 0. By hypothesis, for each e > 0, there exists a compact subset H of G such that \|/(x)\| ^ e for all x ^ H. Take K as in the proof of (ii) above, and suppose that x $ KH. Then if s e K we H and therefore

	JG-K ^ ll/ll '/ which proves that \JL */e

	(14.9.3) £t?ery measure jn on G zj convolvable with every function f e jf~_c(G); r/ze integral on the right-hand side of (14.8.2) is defined for all x e G, awrf /ze function x\-+j f(s~~¹x) d^(s) is continuous on* G. Since the measure/- ft has compact support, p and/* ft are convolvable (14.6.4), and it is clear that the integral tf(s~~¹x) dfj,(s) is defined for all x G G. The continuity of the function x-jf(s~^lx) dfi(s) follows from (14.1.5.5). We shall leave to the reader the task of stating the corresponding proposi- tions for the convolution/* p. It should be noticed in particular that (14.9.2) and its analog for/* /z prove that if G is unimodular (14.3), then «S?g(G) is a left and right module over the algebra M£(G), and the external laws of com- position of these two module structures are compatible by virtue of (14.7.2). (14.9.4) Let p, v be two measures on G, and let fe Jf_c(G). Suppose that ji and v are convolvable. Then the function nfis v-integrable and (14.9.4.1) </,/v> = <Ai/,v>. Likewise, if p and v are bounded and feV$(G), the function /u / is continuous and bounded (hence v-integrable) and the formula (14.9.4.1) is valid.* For the hypothesis implies that the function (s₉ x)\-f(s~^lx)* is (^ ® v> mtegrable, and the result therefore follows from the theorem of Lebesgue- Fubini.