Definition of an integral over linear maps

Question

Let $G$ be a locally compact Hausdorff topological group with Haar measure $\mu$. Let $\varphi \colon G \to GL(V)$ be a continuous representation of $G$, where $V$ is a Hilbert space over $\mathbb C$. Let $C_c(G)$ be the vector space of compactly supported functions $G \to \mathbb C$. Then I've seen the definition $$ \varphi(f) = \int_{g \in G} f(g) \varphi(g) d \mu(g) $$ and I fail to understand what this means and how this integral is defined. I know how an integral over $\mathbb C$-valued functions is defined in measure theory, but now $\varphi(g)$ is a linear map, albeit one over a $\mathbb C$-vector space. Could anyone help me out? Any help is appreciated.