Corollary 2

— Let $G$ be a locally compact group, $A$ and $B$ closed subgroups of $G$ such that $A \supset B$. Suppose that there exists, on the homogeneous space $G/B$ of left cosets with respect to $B$, a nonzero positive measure $\alpha$ that is invariant under $G$ and is bounded.

a) The canonical image of $\alpha$ in $G/A$ is a nonzero positive measure, invariant under $G$, and bounded.

b) $\Delta_G$ coincides with $\Delta_A$ on $A$ and with $\Delta_B$ on $B$.

c) There exists, on the homogeneous space $A/B$ of left cosets of $A$ with respect to $B$, a nonzero positive measure invariant under $A$ and bounded.

The assertion a) is immediate. The assertion b) follows from a) and No. 6, Cor. 2 of Th. 3. By b), $\Delta_A$ coincides with $\Delta_B$ on $B$, and one can therefore apply the results of the present subsection, on taking $X = G$, $H = A$, $H' = B$. The function 1 on $G/B$ is $\alpha$-integrable. By a) of Cor. 1, the function 1 on $A/B$ is integrable for $\beta/\beta'$, where $\beta$ and $\beta'$ denote left Haar measures on $A$ and $B$; thus $\beta/\beta'$ is bounded.