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256 XIV INTEGRATION IN LOCALLY COMPACT GROUPS

virtue of (14.1.5) again, independent of the choice of left Haar measure //. The
mapping s\-+A_G(s) is called the modulus function on G. If/is any /z-integrable
function, the function xt-*f(xs) is therefore also ^-integrable, and we have

f 1 f

(14.3.1.1) f(xs) d/x(*) = A(s" ') /(x)

J J

In particular, if A is any /i-integrable subset of G, then As is also ju-integrable,
and

(1 4.3.1 .2) n(As) = AfcXA).

(14.3.2) The mapping s\-+&_G(s) is a continuous homomorphism ofG into the
multiplicative group R* .

This. follows immediately from the formula (14.3.1.1) and the lemma
(14.1.5.5).

The group G is said to be unimodular if A_G(s) = 1 for all s e G. In this case
there is no distinction to be made between left and right Haar measures, and
we call them simply Haar measures on G.

(14.3.3) If there exists in G a compact neighborhood V of the neutral element e
which is invariant under all inner automorphisms of G, then G is unimodular.
This is the case, in particular, when G is compact, or discrete, or commutative,

For each SE G we have ji(V) = JJL^^VS) = A₀(s()ju(V) by (14.3.1.2) and
(14.1.2.4); hence the result, since /u(V) ^ 0 (14.1.2).

(14.3.4) If u is any left Haar measure on G, then fi = A""¹ • IJL. If f is any
H-integrable function on G, then the function x\-^f(x^^l)^(x)~ ¹ is u-integrable,
and we have

(14.3.4.1)

For each 5- e G we have

5C5XA-¹ • /i) = (50A-¹) • (8(^) = (A^^A-¹) • (A^)^) = A"¹ - u,

which shows that A""¹ • ju is a right Haar measure on G. Since fi is also a right
Haar measure, there exists a constant a > 0 such that ji = aA""¹ • /*. It follows
that n = ^(A"¹ • //T = aA • ji — a²^ (13.14.5), whence a² = 1 and therefore



	256 XIV INTEGRATION IN LOCALLY COMPACT GROUPS virtue of (14.1.5) again, independent of the choice of left Haar measure //. The mapping s\-+A_G(s) is called the modulus function on G. If/is any /z-integrable function, the function xt-f(xs)* is therefore also ^-integrable, and we have

	f 1 f (14.3.1.1) f(xs) d/x(*) = A(s" ') /(x) J J

	In particular, if A is any /i-integrable subset of G, then As is also ju-integrable, and (1 4.3.1 .2) n(As) = AfcXA). (14.3.2) The mapping s\-+&_G(s) is a continuous homomorphism ofG into the multiplicative group R* . This. follows immediately from the formula (14.3.1.1) and the lemma (14.1.5.5). The group G is said to be unimodular if A_G(s) = 1 for all s e G. In this case there is no distinction to be made between left and right Haar measures, and we call them simply Haar measures on G. (14.3.3) If there exists in G a compact neighborhood V of the neutral element e which is invariant under all inner automorphisms of G, then G is unimodular. This is the case, in particular, when G is compact, or discrete, or commutative, For each SE G we have ji(V) = JJL^^VS) = A₀(s()ju(V) by (14.3.1.2) and (14.1.2.4); hence the result, since /u(V) ^ 0 (14.1.2). (14.3.4) If u is any left Haar measure on G, then fi = A""¹ • IJL. If f is any H-integrable function on G, then the function x\-^f(x^^l)^(x)~ ¹ is u-integrable, and we have (14.3.4.1) For each 5- e G we have 5C5XA-¹ • /i) = (50A-¹) • (8(^) = (A^^A-¹) • (A^)^) = A"¹ - u,

	which shows that A""¹ • ju is a right Haar measure on G. Since fi is also a right Haar measure, there exists a constant a > 0 such that ji = aA""¹ • /. It follows that n = ^(A"¹ • //T = aA • ji — a²^* (13.14.5), whence a² = 1 and therefore