Library / Integration / Chapter VII / § 2
Proposition 11
Tag 09G0, permanent. Cite it as [Bourbaki, Tag 09G0].
Integration, chapter VII, § 2. Quotient of a space by a group; homogeneous spaces
— We maintain the notations of Prop. 10. Let u u u G G G u ( G ′ ) = G ′ u(G') = G' u ( G ′ ) = G ′ u ′ u' u ′ u u u G ′ G' G ′ u ′ ′ u'' u ′′ G ′ ′ G'' G ′′ u u u
mod G ( u ) = mod G ′ ( u ′ ) mod G ′ ′ ( u ′ ′ ) .
\operatorname{mod}_G(u) = \operatorname{mod}_{G'}(u') \operatorname{mod}_{G''}(u'').
mod G ( u ) = mod G ′ ( u ′ ) mod G ′′ ( u ′′ ) . For, if α ′ ′ = α / α ′ \alpha'' = \alpha/\alpha' α ′′ = α / α ′ u ′ ′ ( α ′ ′ ) = u ( α ) / u ′ ( α ′ ) u''(\alpha'') = u(\alpha)/u'(\alpha') u ′′ ( α ′′ ) = u ( α ) / u ′ ( α ′ )
mod G ′ ′ ( u ′ ′ ) − 1 α ′ ′ = mod G ( u ) − 1 α / mod G ′ ( u ′ ) − 1 α ′
\operatorname{mod}_{G''}(u'')^{-1}\alpha'' = \operatorname{mod}_G(u)^{-1}\alpha/\operatorname{mod}_{G'}(u')^{-1}\alpha'
mod G ′′ ( u ′′ ) − 1 α ′′ = mod G ( u ) − 1 α / mod G ′ ( u ′ ) − 1 α ′ = mod G ′ ( u ′ ) mod G ( u ) ( α / α ′ ) = mod G ′ ( u ′ ) mod G ( u ) α ′ ′ ,
= \frac{\operatorname{mod}_{G'}(u')}{\operatorname{mod}_G(u)} (\alpha/\alpha') = \frac{\operatorname{mod}_{G'}(u')}{\operatorname{mod}_G(u)} \alpha'',
= mod G ( u ) mod G ′ ( u ′ ) ( α / α ′ ) = mod G ( u ) mod G ′ ( u ′ ) α ′′ , whence the proposition.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Transcribed from the printed volume. Not translated.