Proposition 2

Tag 09EZ, permanent. Cite it as [Bourbaki, Tag 09EZ].

Integration, VII, p. 28, § 2. Quotient of a space by a group; homogeneous spaces

The mapping $f \mapsto f^{\chi}$ of $\mathscr{K}(X)$ into $\mathscr{K}^{\chi}(X)$ is linear, and the image of $\mathscr{K}(X)$ (resp. $\mathscr{K}_{+}(X)$ ) is $\mathscr{K}^{\chi}(X)$ (resp. $\mathscr{K}_{+}^{\chi}(X)$ ).

Linearity is immediate. It is clear that $f^{\chi} \geqslant 0$ for $f \geqslant 0$ . It then suffices to apply the following lemma:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

§ 2. Quotient of a space by a group; homogeneous spaces
Lemma 1, § 2
Proposition 3, § 2
Proposition 6, § 2
Proposition 4, § 5
Proposition 5, § 5

Cites

Nothing.