Integration in a finite dimensional vector space 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.
 A: 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 $\int_G \langle \phi,f(g)\rangle\mu(dg)=\langle \phi,v\rangle$.
A: Definitely do not pick a basis.  You also don't have to pick an inner product on $V$ or reduce to scalar-valued functions.
You can define integration with respect to a measure in direct way as a limit without having to do any kind of decomposition into real or imaginary parts or positive or negative parts as is often presented for integration of real or complex valued functions.  See the treatment of integration in Lang's "Real and Functional Analysis", where the functions being integrated take values in a Banach space. That applies in particular to functions with the values in a finite-dimensional complex vector space $V$ (which has a canonical Hausdorff topology making it complete for all vector space norms on $V$).
