Corollary 2

— Let $\lambda$ be a positive measure on $X/H$, and $f$ a $\lambda^\#$-measurable numerical function on $X$. Suppose that, for every $\xi \in H$, $\delta(\xi)f = f$ locally $\lambda^\#$-almost everywhere. Then, there exists a $\lambda$-measurable function $g$ on $X/H$ such that $f = g \circ \pi$ locally $\lambda^\#$-almost everywhere.

Replacing $f$ by $f/(1+|f|)$, one reduces to the case that $f$ is bounded, hence locally $\lambda^\#$-integrable. Let $\mu = f \cdot \lambda^\#$. The hypothesis on $f$ implies that $\delta(\xi)\mu = f \cdot \delta(\xi)\lambda^\# = \Delta_H(\xi)\mu$ for all $\xi \in H$. There then exists (Prop. 4) a measure $\lambda'$ on $X/H$ such that $\mu = {\lambda'}^\#$. By Cor. 1, there exists a locally $\lambda$-integrable function $g$ on $X/H$ such that $\lambda' = g \cdot \lambda$. By Prop. 6, $f \cdot \lambda^\# = {\lambda'}^\# = (g \circ \pi) \cdot \lambda^\#$, whence $f = g \circ \pi$ locally $\lambda^\#$-almost everywhere.