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
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