Definition of the set of independent r.v. with second moment contstraint

Question

I am trying to nice write the definition of the following set.

Def: The set of all distributing of the pair $(X_1,X_2)$ such that

1. $X_1$ and $X_2$ are independent

2. Have second moment constraint $E_{F(X_1)}[X_1^2] \le 1$, $E_{F(X_2)}[X_2^2] \le 1$

I thought of the following: \begin{align*} S_1&=\left\{F(X_1,X_2)\Big| F(X_1,X_2)=F(X_1)F(X_2) \text{ and } E_{F(X_1)}[X_1^2] \le 1, \ E_{F(X_2)}[X_2^2] \le 1 \right\}\\ S_2&=\left\{(F(X_1),F(X_2) )\Big| F(X_1,X_2)=F(X_1)F(X_2) \text{ and } E[X_1^2] \le 1, \ E[X_2^2] \le 1 \right\} \end{align*}

Which on is more correct $S_1$ or $S_2$? Also, should I write $E_{F(X_1)}[X_1^2]$ or $E_{F(X_1,X_2)}[X_1^2]$ or it doesn't really matter? Thanks you for any help and comments.

Answer 1

This is the set $S$ of product distributions whose marginals both have second moment at most $1$. $$S=\left\{\mu\otimes\nu\in\mathcal M_1^+(\mathbb R)\times\mathcal M_1^+(\mathbb R)\,\left|\,\int_\mathbb R x^2\mathrm d\mu(x)\leqslant1,\,\int_\mathbb R x^2\mathrm d\nu(x)\leqslant1\right.\right\}$$