Let $V$ be a finite dimensional complex vector space. Let $G$ be a compact group with normalized Haar measure $\mu$. In the representation theory of compact groups, I encounter


$$\int\_G f(g) \mu(dg)$$


where $f: G \to V$ is a continuous function.


How is this integral definied? I know about the Lebesgue integral for functions with values in $\mathbb{C}$.


Maybe we should fix a base and consider the isomorphism $V \cong \mathbb{C}^n$ and apply the integral in each component? But then it should be shown that this does not depend on the choice of basis. Moreover, choosing a basis is always non-canonical so I might be missing something here.



Integration in a finite dimensional vector space
If you do not want to use coordinate systems, a more intrinsic way to reduce to scalar functions is defining $\int\_Gf(g) \mu(dg)$ as the (unique) element $v\in V$ such that for any linear form $\phi\in V^\*$ one has $\int\_G \langle \phi,f(g)\rangle\mu(dg)=\langle \phi,v\rangle$.


