I'm trying to verify formula (5.1) in Getz's and Hahn's An Introduction to Automorphic Representations, §5.2, p. 138. The goal is to calculate the product of basis elements in the Hecke algebra.
The setting is as follows. Let $G$ be (for simplicity) a locally profinite group, and let $K \le G$ be a compact open subgroup. Choose a right Haar measure $\mu$ on $G$. For any $a,b \in G$, we want to calculate the convolution $$1_{KaK} * 1_{KbK}(g) := \int_G 1_{KaK}(gh^{-1}) 1_{KbK}(h) dh$$ where $1_X$ is the characteristic function on a set $X$.
The approach is as follows: since $KaK$ and $KbK$ are compact, write $KaK = \coprod_i Ka_i$ and $KbK = \coprod_j b_jK$ as a finite disjoint union where $a_i, b_j \in G$. We then obtain $$\int_G 1_{KaK}(gh^{-1}) 1_{KbK}(h) dh = \sum_{i,j} \int_G 1_{Ka_i}(gh^{-1}) 1_{b_jK}(h) dh.$$ The integrand is one iff $h \in a_i^{-1}Kg \cap b_jK$. Hence $$\sum_{i,j} \int_G 1_{Ka_i}(gh^{-1}) 1_{b_jK}(h) dh = \sum_{i,j} \mu(a_i^{-1}Kg \cap b_jK).$$ One now checks that $a_i^{-1}Kg \cap b_jK$ is non-empty iff $g \in Ka_ib_jK$. By right $G$-invariance of $\mu$, we thus have $$\sum_{i,j} \mu(a_i^{-1}Kg \cap b_jK) = \sum_{i,j} \mu(a_i^{-1}Ka_ib_j \cap b_jK) \cdot 1_{Ka_ib_jK}(g).$$
At this point, I'm stuck. How do Getz-Hahn deduce that $$\sum_{i,j} \mu(a_i^{-1}Ka_ib_j \cap b_jK) \cdot 1_{Ka_ib_jK}(g) = \sum_{i,j} \mu(b_jK) \cdot 1_{Ka_ib_jK}(g)?$$ I think that $\mu(a_i^{-1}Ka_ib_j \cap b_jK) = \mu(b_jK)$, which is the case if $b_jK \subseteq a_i^{-1}Ka_ib_j$.
