I am currently reading up on measure theory and Lebesgue integration in order to gain a somewhat more nuanced understanding of a few "aggregation-adjacent" results in economics.
The context of my question essentially is that we have a continuum of agents collected in a set $H$ with Lebesgue measure 1.
Each of these agents $i\in H$ faces an individual price $p_i$ such that we basically have a function $p:H\to\mathbb{R}$. Agent $i$ also consumes a quantity given by $x:H\to\mathbb{R}$ and written as $x_i$.
Clearly the cost of consumption for each agent is given by $p_i\,x_i$ which yields an aggregate (and by the measure unity assumption average) cost of $$\int_Hp_i\,x_i\,di.$$
So far, perfectly reasonable.
At this point a variety of textbooks just states - in a very cavalier way - that $$\dfrac{\partial}{\partial\,x_k}\int_Hp_i\,x_i\,di=p_k.$$
Why is that true?
I suppose that, provided $H$ was countable, we would think about the integral as a sum. And clearly, $$\dfrac{\partial}{\partial\,x_k}\sum_{i\in H}p_i\,x_i=p_k$$
However, $H$ is a continuum and we integrate w.r.t. to the usual Lebesgue measure.
I'd very much appreciate is someone could give me a rigorous argument for why the above is true and perhaps also an intuition for how I should think about this. After all, we are not differentiating w.r.t. a parameter such that we'd need some sort of Dominated Convergence Theorem type of argument, we are not differentiating in a Radon-Nikodym way. We are essentially differentiating the integral of a function $i\mapsto p_ix_i$ over $H$ with respect to a "component" of this function evaluated at a specific point in the domain of this function, i.e. $x(k)$. Not quite sure how to make sense of this.
Thank you very much.
Consider taking the directional derivative in the direction of the $\delta_k$ function:
$$ \lim_{\tau \rightarrow 0} \dfrac{F(x+\tau \delta_k )-F(x)}{\tau} =\lim_{\tau \rightarrow 0} \dfrac{\int_0^1 p_i (x_i+\tau \delta_k)d_i - \int_0^1 p_i x_i d_i}{\tau} = \lim_{\tau \rightarrow 0} \dfrac{\int_0^1 \tau p_i \delta_k di}{\tau} = \int_0^1 \delta_k p_i d_i = p_k. $$
I find it to be true that economists almost always mean some kind of directional derivative.