I have a question regarding how an integral is defined in the following case. If we consider the real vector space $\mathcal{M}^{m \times n}$ of $m \times n$ matrices equipped with an inner product. Consider a function $f: \mathcal{M}^{m \times n} \rightarrow \mathcal{M}^{m \times n}$. How is the following integral usually defined $$\int_{\mathcal{M}}f(\lambda) d \nu(\lambda)$$
Thanks for any help.
As a first approach you can do this component wise. The trouble with this is that usually you want to have a definition which does not depend on the choice of a basis, so this is what you would have to check in this case.
A (very) general definition can be found in several places, e.g. in Rudin's functional analysis, chapter 3, section 'Vector valued integration'. It requires for a function with values in a topological vector space $X$ that its integral commutes with continuous linear maps, i.e. the equation $$ \Lambda \left(\int_X f d\nu \right) = \int_X \Lambda(f) d\nu$$ has to hold for any $\Lambda \in X^*$. To see that this is well defined have a look at Rudin's book.
Or google 'vector valued integration'.