Number Theory: An Introduction to Mathematics

8 Generalizations 433

The notions of left and right Haar integral obviously coincide if the groupGis
abelian, and it may be shown that they also coincide ifGis compact or is a semi-
simple Lie group.
We now restrict attention to the case of a left Haar integral. It is easily seen that

M(f ̄)=M(f),

wheref ̄(s)=f(s)for everys∈G.Ifweset(f,g)=M(fg ̄), then the usual inner
product properties hold:

(f 1 +f 2 ,g)=(f 1 ,g)+(f 2 ,g),

(λf,g)=λ(f,g),

(f,g)=(g,f),

(f,f)≥ 0 ,with equality only iff≡ 0.

By theRiesz representation theorem, there is a uniquepositive measureμon the
σ-algebraMgenerated by the compact subsets ofG(cf. Chapter XI,§3) such that
μ(K)is finite for every compact setK⊆G,μ(E)is the supremum ofμ(K)over all
compactK⊆Efor eachE∈M,and

M(f)=

∫

G

fdμ for everyf∈C 0 (G).

The measureμis necessarily left invariant:

μ(E)=μ(sE) for allE∈Mands∈G,

wheresE={sx:x∈E}.
Forp=1or2,letLp(G)denote the set of allμ-measurable functionsf:G→C
such that
∫

G

|f|pdμ<∞.

The definition ofMcan be extended toL^1 (G)by setting

M(f)=

∫

G

fdμ,

and the inner product can be extended toL^2 (G)by setting

(f,g)=

∫

G

fg ̄dμ.

Moreover, with this inner productL^2 (G)is aHilbert space.Ifwedefinetheconvolu-
tion product f∗goff,g∈L^1 (G)by

f∗g(s)=

∫

G

f(st)g(t−^1 )dμ(t),