Let $(G,M,\mu)$ be a measure space, where $\mu$ is the Haar measure on topological group $G:=\mathbb R\times\mathbb R_d$, ($\mathbb R$ is the group of reals with the natural topology whereas $\mathbb R_d$ is the group of reals with the discrete topology) and $M$ be the $\sigma$-algebra of all Haar measurable subsets of $G$.

Let $\mu_0 :=\mu|_B$, where $B$ is the $\sigma$-algebra of all Borel subsets of $G$, and let $(G,M_1,\mu_1)$ be the smallest completion of the measure space $(G,B,\mu_0)$.

Is it true that $M_1=M$ and consequently $\mu_1=\mu$ ?