Difference Between Product and Function Spaces This confusion arose in the context of thinking about different ways to interpret stochastic processes.

Say we have an index set $J$ and some other space $X$.
Then is there a way to formalize the intuition that "$X \times J \simeq X^J$"?

$X^J$ can represent all functions $J \to X$, but at the same time every function $J \to X$ can also be represented as a subset of $X \times J$ if I am thinking correctly.
So it is all of the sudden unclear to me how to satisfactorily differentiate them on a conceptual level, which is alarming to me, since their set-theoretic definitions are clearly extremely different (one is a finite Cartesian product and the other is possibly an infinite Cartesian product).
We use such a relationship all of the time, for example, when we draw functions $f: \mathbb{R} \to \mathbb{R}$ in $\mathbb{R}^2$.
One can also identify vectors in $\mathbb{R}^n$ with functions $\{1,\dots,n\} \to \mathbb{R}$.
Since any function $h: J \to X$, being a relation, is a subset of $X \times J$, it must therefore also be an element of the power set, i.e. $h \in \mathcal{P}(X \times J)$.

So is it true then that $X^J \subset \mathcal{P}(X \times J)$?

Please let me know if what I am asking is unclear so I can improve the question. Right now I am confused enough that I am not sure I am correctly explaining the source of my confusion.
Something similar to this is mentioned in this Math.SE answer: https://math.stackexchange.com/a/312111/327486. However, the answer only adds to my confusion about this identification, rather than alleviating it.
 A: I prefer to write ${^JX}$ for the set of functions from $J$ to $X$. Each $f\in{^JX}$ is, as you say, a relation, but not from $X$ to $J$: it’s a relation from $J$ to $X$, so ${^JX}\subseteq\wp(J\times X)$. This means that if $f\in{^JX}$, then $f\in\wp(J\times X)$, and therefore $f\subseteq J\times X$.
You write:

So it is all of the sudden unclear to me how to satisfactorily differentiate them on a conceptual level, which is alarming to me, since their set-theoretic definitions are clearly extremely different (one is a finite Cartesian product and the other is possibly an infinite Cartesian product).

No, one is a possibly infinite Cartesian product, and the other is a subset of a finite Cartesian product; there is no conflict here at all. ${^JX}$ contains precisely those subsets of $J\times X$ that (a) are functions, and (b) have domain $J$. ($J\times X$ also has subsets that are partial functions, i.e., functions whose domains are proper subsets of $J$.)
A: First if all, I think using "map" instead of "function" in this situation is more appropriate. 
Now for any $x\in X$, one can associated any element $j\in J$. This gives us a pair of $(x,j)\in X\times J$. On the other hand, this also define an association in a map which sending $x\mapsto j$. Therefore, this two set can be consider to have the same structure. One can first use the finite sets as an example, then it should not be obscure. 
