Consider the set $Y = \{X_1,\cdots,X_4\}$ where $X_1,\cdots,X_4$ are identically distributed but not necessarily independent random variables. The random variable $X_1$ (any $X_i, i \in [4]$), can take $n$ distinct values, say $a_1,\cdots,a_n$.
The set of possible distributions of $X_1$, $\Lambda = \{\alpha \in \mathbb{R}^n: \alpha \geq 0, \sum\limits_{j \in [n]} \alpha_j = 1\}$.
The set of possible distributions of $Y$, $\Gamma = \{\mu\in \mathbb{R}^{\overline{n}}: \mu \geq 0, \sum\limits_{j \in [\overline{n}]} \mu_j = 1\}$, where $\overline{n}$ is the number of possible "values" the set $Y$ can take. (I use sets instead of vectors because order is not important, i.e. $(a_1,a_2,a_3,a_4)$ is considered the same as $(a_2,a_4,a_3,a_1)$).
The set of possible distributions of $Y$ when $X_i$'s are independent, $\Gamma^I = \{\mu\in \mathbb{R}^{\overline{n}}: \mu(\{b_1,b_2,b_3,b_4\}) = \prod\limits_{j \in [4]}\alpha(b_j)$, where $b_j \in \{a_1,\cdots,a_n\}$ for all $j \in [4]$.
My question is, is the set $\Gamma^I$ dense in the set $\Gamma$?
I can see that $\Gamma^I$ is a measure zero set with respect to $\Gamma$ but not sure how to go about determining if it's dense or not.
PS: I have not tagged probability related topics because while the setting involves random variables, the question is purely related to denseness in Euclidean spaces.
Write $\Omega = \{a_1,\ldots,a_n\}^4$. If $\mu \in \Gamma \backslash \Gamma^I$, there is some $b = (b_1,b_2,b_3,b_4) \in \Omega$ such that $\mu((b_1, b_2, b_3, b_4)) \ne \prod_j \mu(\{x \in \Omega: x_j = b_j\})$. This inequality will still be true for $\mu'$ if $\sum_{b\in\Omega} |\mu(b) - \mu'(b)|$ is sufficiently small. Therefore it is not dense.