Expectation value - does it need to be on a Banach space? I came across the following definition of a exception value of a random variable $X$ in (Lord, Powell and Shardlow, 2014;pg139; emphasis not mine):

Let $X$ be a Banach space-valued random variable on the probability
  space $(\Omega,\mathcal{F},\Bbb{P})$. If $X$ is integrable, the
  expectation of the $X$ is $$\Bbb{E}[X]:= \int_\Omega X(\omega)d\Bbb{P}(\omega), \tag{4.1}$$

Why is there a resection to Banach space-random variables here? Why won't only old measurable space do?
 A: There are two objects being considered here - the sample space $\Omega$, and an unnamed state space which I shall call $E$, which in this case is assumed to be a Banach space. (It is fairly unhelpful that the author does not give the state space a name, and probably contributes to your confusion.) Our random variable is $X:\Omega\to E$.
Random variables need not take values in Banach spaces. Indeed, at its most basic definition, a random variable is just a measurable map from a probability space to a measurable space. However, for almost all practical purposes, we require some topology on the state space. Do you want any kind of notion of convergence? Then you need some kind of topology.
(Edit, to make something clear from the comments: whenever you have a topological space, you also have a measurable space by considering the Borel $\sigma$-field. Unless otherwise specified, you should always assume this is the $\sigma$-field considered when we have a topological space being considered in measure/probability theory.)
In this case, we are looking at the expectation of a random variable, which we know is simply the integral of the measurable function $X$. But what is the integral? Any definition of an integral requires linear combinations and limits, so at the least we will require $E$ to be a topological vector space. Assuming $E$ is Banach simply makes things a little cleaner - the topology of a normed vector space is easier to understand, and making it complete (i.e. Banach) merely guarantees that the space possesses enough limits to avoid running into problems.
