Lebesgue integral of vector-valued function? In Bernt Øksendals stochastic differential equations he says that if we have a random variable $X:\Omega\rightarrow\mathbb{R}^d$. He defines the expectation:
$E[X]=\int_\Omega X(\omega)dP(\Omega)=\int_{\mathbb{R}^d}xd\mu_X(x)$.
But what is the definition of an integral over a vector-valued function? I've tried to find it in various measure theory books, but the most I could find is integration of complex valued function. How is this integral defined, have you any references for measure or probability books, where this is defined? It seems likely that we integrate componentwise in some way, but I'd like to see some rigorous way of defining this integral so that we know everything works.

He is probably referring to the Bochner integral.
Edit: The Wikipedia page does a good job of presenting the definition. To answer your question in the comments, there is a natural theory of vector-valued measures, but when the vector space is $\mathbb{R}^d$ (and other finite products of measure spaces) these are just tuples $\mu = (\mu_1,\ldots,\mu_d)$ of measures $\mu_i$. Integration against $\mu$ is done component-wise.
