Consumer optimum in an economy with a continuum of commodities Consider an economy with a continuum of commodities, with one commodity for each point in $[0,1]$.

Suppose a consumer wants to maximise
$$U = \int_0^1 c_i^\theta\,di\qquad 0&lt;\theta&lt;1$$
subject to
$$\int_0^1 p_i c_i\,di = M$$
where $c_i$ is the amount of the $i$-th commodity consumed, $p_i$ its price and $M$ the consumer's money income.

This kind of problem arises for example in applying the Dixit-Stiglitz model to macroeconomics or international trade.

The solution to this problem is supposedly
$$c_i = Ap_i^{1 \over {\theta-1}}$$
where $A$ is a constant chosen to ensure that the budget constraint is satisfied.

I am not very satisfied with derivations of this result which use Lagrange multipliers in analogy with the case of a finite number of commodities. What would be a completely mathematically rigorous method of deriving the above result?

It seems clear that there isn't a unique solution since arbitrarily changing the values of $c_i$ for a finite number of values of $i$ will leave the integrals in the utility function and budget constraint unchanged. I am expecting that a completely rigorous derivation would also correctly pinpoint this degree of nonuniqueness.

EDIT: In response to the comments by @BKay, @Ubiquitous. My problem with starting out with economies with $n$ commodities and taking the limit as $n \to \infty$ is that this needs to be accompanied by an argument which shows that the limit of optima is an optimum of the limit problem. I would appreciate a reference to a result which shows this either for this particular problem or a general result which is applicable to this problem.

In response to @AlecosPapadopoulos. The proofs of the Langrange multiplier method that is taught in math for economics courses is usually for a finite number of choice variables. I would appreciate a reference to where the method is justified for a continuum of choice variables. Also, the nonuniqueness I mention above shows that the method cannot be exactly right. Then what exactly are the qualifications required for its validity?
As the OP noted in a comment, Theorem 1 in section 12 of Kolomogorov and Fomin's Calculus of Variations seems to provide some comfort that we can indeed use the Langrange Multiplier method when the number of our variables is infinite. Still, the authors do that in a footnote, writing "the reader will easily recognize the analogy with Langrange multipliers". So no, this does not rigorously show what we want.  

I think that what we need is a paper like Craven, B. D. (1970). A generalization of Lagrange multipliers. Bulletin of the Australian Mathematical Society, 3(03), 353-362. which in its summary writes:


  The method of Lagrange multipliers for solving a constrained
  stationary-value problem is generalized to allow the functions to take
  values in arbitrary Banach spaces (over the real field). The set of
  Lagrange multipliers in a finite-dimensional problem is shown to be
  replaced by a continuous linear mapping between the relevant Banach
  spaces.


This is math-speak but it says what we wanted to hear (one can also find a short exposition in wikipedia to the degree that it trusts the content).  

Then, we can form the Lagrangean of the problem

$$\Lambda = \int_0^1 c_i^\theta\,\text{d}i +\lambda\left(M - \int_0^1 p_i c_i\,\text{d}i\right)$$

and calculate the first-order condition(s) by , informally speaking, "looking at the integral and seeing a sum",

$$\frac{\partial \Lambda}{\partial c_i} = 0 \Rightarrow  \theta c_i^{\theta-1} = \lambda p_i, \;\; i \in [0,1] \tag{1}$$

...a continuum of conditions. For later use we  define

$$\sigma \equiv 1/(1-\theta), \Rightarrow 1-\theta = 1/\sigma, \;\; \theta = \frac {\sigma-1}{\sigma}$$

The constant $\sigma$ can be shown to be the elasticity of substitution between any two goods.

Writing $(1)$ for commodity $j$ and equating through the common lagrange multiplier we arrive at 

$$c_i = \left(\frac {p_i}{p_j}\right)^{-\sigma}c_j
 \tag{2}$$

Multiply both sides by $p_i$ and take the integral over the commmodity space with respect to $i$:

$$\int_0^1 p_i c_i\,\text{d}i = \int_0^1 p_i^{1-\sigma}p_j^{-\sigma}c_j\,\text{d}i$$

$$\Rightarrow M = p_j^{\sigma}c_j\int_0^1 p_i^{1-\sigma}\,\text{d}i$$

$$\Rightarrow c_j = p_j^{-\sigma} \cdot M \cdot \left(\int_0^1 p_i^{1-\sigma}\,\text{d}i\right)^{-1} \tag{3}$$

which is Marshallian demand for commodity $j$.