Let $G$ be a locally profinite group (i.e. a topological group that is locally compact Hausdorff and totally disconnected or, equivalently, a Hausdorff topological group s.t. $1 \in G$ has a neighbourhood basis consisting of compact open subgroups). Let $K \subset G$ be a compact open subgroup and denote by $C_c^\infty(G // K)$ the set of locally constant functions $f : G \to \mathbb C$ s.t. $f(k_1gk_2) = f(g)$ for all $g \in G$ and $k_1, k_2 \in K$. Assume that $G$ is unimodular and fix a Haar measure $\mu$ on $G$ a. For $f_1,f_2 \in C_c^\infty(G // K)$ define the convolution $f_1 * f_2 \in C_c^\infty(G // K)$ by \begin{align*} (f_1 * f_2)(g) := \int_G f_1(gx^{-1})f_2(x) d\mu(x). \end{align*}
Let $\alpha, \beta \in G$ and write \begin{align*} K\alpha K &= \bigsqcup_{i = 1}^n K\alpha_i, \\ K \beta K &= \bigsqcup_{j = 1}^m \beta_j K. \end{align*} Then it is claimed on page 136 in [GH] that the convolution of the two indicator functions $\mathbb{1}_{K\alpha K}$, $\mathbb{1}_{K \beta K}$ is given by \begin{align} \mathbb{1}_{K\alpha K} * \mathbb{1}_{K \beta K} = \sum_{i,j} \mu(\beta_jK) \mathbb{1}_{K\alpha_i \beta_j K}. \end{align}
Question: Why is that true?
My attempt: Since $gx^{-1} \in K \alpha K$ is equivalent to $x \in K \alpha^{-1} Kg$, I get \begin{align*} (\mathbb{1}_{K\alpha K} * \mathbb{1}_{K \beta K})(g) &= \mu(K \alpha^{-1} Kg \cap K\beta K) \\ &= \sum_{i,j} \mu(\alpha_i^{-1}Kg \cap \beta_j K) \end{align*} Now $\alpha_i^{-1}Kg \cap \beta_j K \neq \emptyset$ is equivalent to $g \in K\alpha_i \beta_j K$. Hence, I get \begin{align*} \mathbb{1}_{K\alpha K} * \mathbb{1}_{K \beta K} = \sum_{i,j} c_{i,j} \mathbb{1}_{K\alpha_i\beta_j K} \end{align*} with \begin{align*} c_{i,j} = \mu(\alpha_i^{-1}Kg_{i,j} \cap \beta_j K) \end{align*} for any $g_{i,j} \in K\alpha_i\beta_j K$ (e.g. $g_{i,j} = \alpha_i\beta_j$). But \begin{align*} \mu(\alpha_i^{-1}Kg_{i,j} \cap \beta_j K) < \mu(\beta_j K) \end{align*} in general, hence (1) does not hold. Where did I make a mistake?
Reference: [GH] Jayce R. Getz, Heekyoung Hahn. An Introduction to Automorphic Representations (version that had been available on the second author's webpage until recently)