function with a continuum of inputs (economic application) In economics, we often use real-valued functions of the following type:
$$U (x_1, x_2)$$
$x_1$ and $x_2$ are the quantities of two goods (real numbers). It is straightforward to work with this kind of function with an arbitrarily many finite or countably infinite number of goods instead of just two. However, I am currently doing some research that could benefit from allowing a continuum of goods (economics is realistic like that). For the type of application I am working on, it is common to assume a continuum of goods with the measure normalized to $1$. 
In existing applications involving a continuum of goods, people always (to my knowledge) assume that $U$ is additively separable and so can be written as an integral. I do not want to assume additive separability, however. [Just in case additive separability is a term only economists use, in the two-good example above, it means that $U (x_1, x_2)$ can be written as $f(x_1)+g(x_2)$.]
I was wondering what the type of function I am looking for is called (if it has a specific name) and if there are references that deal with this type of object. More specifically, I wanted to know the notation to be used in writing down such a function and also see generalizations of standard calculus results (e.g. the chain rule) that are applicable to these cases.
Thanks!
 A: What you can do is take, for the space of bundles, the space of signed measures on, say, $[0,1]$ with the Borel $\sigma$-algebra (this set has Lebesgue measure $1$). The utility function can then be a linear (or concave) functional on this space of measures. This generalizes the finite-good case. An element of $\mathbb{R}^n$ is a signed measure on the finite set $\{ 1,\cdots, n\}$.
If your utility is linear in the bundles, i.e. goods are perfect substitutes, then it can simply be a measurable function $f$ on $[0,1]$: if $\mu$ is a bundle,
$$
U(\mu) = \int f d \mu.
$$
Perfect substitutes on $\mathbb{R}^n$ is again a special case. This is essentially what is done in consumer theory with uncertainty. There the bundles are probability measures/lotteries on the set of outcomes and $f$ is the von Neumann-Morgentern utility. (Note that $f$ being concave, i.e. agent risk-averse, is not the same as the utility function on lotteries being concave. Von Neumann-Morgenstern utility is linear in lotteries.) 
In the linear case, the appropriate notion of the derivative should tell you that, if a utility function is represented by a continuous $f$, the marginal utility of good $x \in [0,1]$ should be $f(x)$.
If utility is concave (for a convex preference), then it would help to know the specific context. But in general it's not difficult to cook up concave functionals from linear ones. 
