Supposing $Y_1,Y_2,\cdots, Y_n$ be random variables such that $Y_i \in \mathcal{L}^2(\Omega,\Sigma,P)$ for all $i$. What are the conditions under which $\mathrm{span}(Y_1,Y_2,\cdots,Y_n) = \mathcal{L}^2(\Omega,\sigma(Y_1,Y_2,\cdots,Y_n),P) ?$
I think that this should be true when $Y_1,Y_2,\cdots,Y_n$ are jointly Gaussian, but even in this case, a proof would be highly appreciated.
Thanks, Phanindra