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2 PARTICULAR CASES AND EXAMPLES 253

We shall use this local definition of a Haar measure in Chapter XIX to
construct a left Haar measure on a Lie group. Here we note the following
consequence of (14.2.4):

(14.2.5) Let G be a locally compact group, H a discrete normal subgroup
of G, and n : G -* G/H the canonical homomorphism. Also let V be an open
neighborhood of the neutral element of G such that the restriction of n to V is
ahomeomorphism of V onto the neighborhood n(V) of the neutral element
of G/H (12.11.2). Let A be a left Haar measure on G. If n is the image under
n | V of the restriction l_v of 1 to V, then ju is the restriction to n(V) of a left
Haar measure on G/H.

For every open set in n(V) is of the form 7r(U), where U c V is open, and
the relation n(s)n(U) an(V) is equivalent to sU<=V; hence it follows
immediately from the definitions that ju satisfies the condition of (14.2.4).

Example

(14.2.6) The mapping cp : t\-+e^2Klt is a strict morphism (12.12.7) of R onto
the compact group U of complex numbers of absolute value 1, by virtue of
(9.5.2) and (9.5.7). The kernel of q> is the discrete subgroup Z consisting of the
integers, and U may therefore be canonically identified with the quotient
group R/Z = T (also called the \-dimensional torus or the additive group of
real numbers modulo 1). Apply (14.2.5) to the case where V = ] —i, i[;
bearing in mind that a Haar measure \JL on U must be diffuse (14.2.3) and that
the complement of <p(V) in U consists of a single point, we see that a function
/on U is ju-integrable if and only if the function t\-~*f(e^2ltit) is Lebesgue-

integrable on ]-i, if, and that we then have \fd\t. = f* ² f(e²***) dt.

J J —1/2

(14.2.7) Let G! , G₂ be two locally compact groups, and ^ (resp. u₂) a left
Haar measure on Gj (resp. G₂). Then /^ ®/^₂ ^JS ^a left Haar measure on
G_t x G₂.

For each function/e JT_C(G! x G₂) and each (s_l9 s₂) e G_t x G₂ we have

= dp^xi) f(s_lx^



	2 PARTICULAR CASES AND EXAMPLES 253 We shall use this local definition of a Haar measure in Chapter XIX to construct a left Haar measure on a Lie group. Here we note the following consequence of (14.2.4): (14.2.5) Let G be a locally compact group, H a discrete normal subgroup of G, and n : G -* G/H the canonical homomorphism. Also let V be an open neighborhood of the neutral element of G such that the restriction of n to V is ahomeomorphism of V onto the neighborhood n(V) of the neutral element of G/H (12.11.2). Let A be a left Haar measure on G. If n is the image under n \| V of the restriction l_v of 1 to V, then ju is the restriction to n(V) of a left Haar measure on G/H. For every open set in n(V) is of the form 7r(U), where U c V is open, and the relation n(s)n(U) an(V) is equivalent to sU<=V; hence it follows immediately from the definitions that ju satisfies the condition of (14.2.4).

	Example (14.2.6) The mapping cp : t\-+e^2Klt is a strict morphism (12.12.7) of R onto the compact group U of complex numbers of absolute value 1, by virtue of (9.5.2) and (9.5.7). The kernel of q> is the discrete subgroup Z consisting of the integers, and U may therefore be canonically identified with the quotient group R/Z = T (also called the \-dimensional torus or the additive group of real numbers modulo 1). Apply (14.2.5) to the case where V = ] —i, i[; bearing in mind that a Haar measure \JL on U must be diffuse (14.2.3) and that the complement of <p(V) in U consists of a single point, we see that a function /on U is ju-integrable if and only if the function t\-~f(e^2ltit)* is Lebesgue- integrable on ]-i, if, and that we then have \fd\t. = f* ² f(e²*) dt. J J —1/2 (14.2.7) Let G! , G₂ be two locally compact groups, and ^ (resp. u₂) a left Haar measure on Gj (resp. G₂). Then /^ ®/^₂ ^JS ^a left Haar measure on G_t x G₂. For each function/e JT_C(G! x G₂) and each (s_l9 s₂) e G_t x G₂ we have

	JI

	= dp^xi) f(s_lx^