Direct limit of $L^2$ spaces

Question

Let $G$ be a discrete group and consider all the subgroups $N$ of finite index, these form a directed as follows: if $N \subseteq N'$ (in this case we say that $N' < N$) there is a morphism $i_{N',N}:G/N \to G/N'$ given by projection. The projective limit $\widehat{G} = \varprojlim_N G/N$ exists and it is called the profinite completion of $G$. On each quotient $G/N$ there is a normalized Haar measure $\mu_N$ and the same is true for $\widehat G$ with a measure $\mu$. It is true that $$L^2(\widehat{G},\mu) \cong \varinjlim_N L^2(G/N,\mu_N)$$ where the morphism ${i_{N',N}}^*:L^2(G/N') \to L^2(G/N)$ is given by composition with $i_{N',N}$?