14

I. HECKE L-FUNCTIONS

IT dfiv x d\xw x

djji1

ves

= IT dfiy x dfj?

veT

= dfiT,

which is what we wanted to prove. We thus have that the measures {d/j,s} form a

coherent family and determine a unique measure on the restricted direct product

which we call product measure and which will be denoted by

dn = Y\dVv

A basis for the family of measurable sets is given by the subsets of G of the

form

o=n^

where each Ov is a measurable subset of Gv with finite d[xv-measure and each

Ov — Hv for almost all v.

The limit of a sequence as with values in a topological space, indexed by finite

subsets S is denoted lim^a^ = a0 and as usual means the following: given any

neighborhood j\f of ao there exists a subset Sjsf

s u c n

that as € j\f for all S D S//.

The integral of a function 3 : G — C, denoted by

/ $(a)d/i(o),

JG

is defined in analogy with the concept of Lebesgue sum as the value of the integrals

fc $(a)da as the set C runs over larger and larger compact subsets of G. Since any

such compact subset C is contained in some G5, we therefore conclude that the

integral of a measurable function makes sense and is definable as a limit

/ f(a)dii(a) =: lim / ^(a)d/jJ(a),

JG S JGs

IG ° JG

S

whenever one of the following two conditions is satisfied:

(i) $(0) 0, in which case the value +00 is allowed

(ii) |$(a)| has a finite integral.

As usual the measurable functions $ satisfying condition (ii) above form a

linear space denoted by L\{G) with the analogous definition for L\(GV).

We introduce a family of functions on the restricted direct product for which

the associated integral has a representation by an "Euler-type" infinite product.

For each index v let $v : Gv —• C be a continuous complex valued function in