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Let $\{v_\lambda:~\lambda\in\Lambda\}$ be a family of real-valued random variables on a (complete) probability space $(\Omega, \sigma, \mathbb{P})$. Assume the variables lie in $\bigcap_{p=1}^\infty L^p(\Omega)$ and denote by $\mathcal{A}$ the unital algebra generated by them along with the constants, and $\alpha$ the minimal complete sub-sigma-algebra with respect to which they are measurable.

If the variables are Gaussian then it is a consequence of the Wiener chaos decomposition (see e.g. Janson - Gaussian Hilbert Spaces) that $\mathcal{A}$ is dense in $L^2(\Omega, \alpha, \mathbb{P}|_\alpha)$. In particular, $L^2(\Omega,\alpha,\mathbb{P}|_\alpha)$ is generated by $\mathcal{A}$ as a subspace of $L^2(\Omega, \sigma, \mathbb{P})$.

But what can we say in the general case? Is it possible that the algebra and $\sigma$-algebra generate different subspaces? Relevant references would be appreciated - mostly I'm having trouble searching for literature about actual algebras of random variables because the term '$\sigma$-algebra' appears almost everywhere.

Let $\{v_\lambda:~\lambda\in\Lambda\}$ be a family of real-valued random variables on a (complete) probability space $(\Omega, \sigma, \mathbb{P})$. Assume the variables lie in $\bigcap_{p=1}^\infty L^p(\Omega)$ and denote by $\mathcal{A}$ the unital algebra generated by them (along with the constants), and $\alpha$ the $\sigma$-algebra generated by them (i.e. the minimal complete sub-sigma-algebra with respect to which they are measurable).

If the variables are Gaussian then it is a consequence of the Wiener chaos decomposition (see e.g. Janson - Gaussian Hilbert Spaces) that $\mathcal{A}$ is dense in $L^2(\Omega, \alpha, \mathbb{P}|_\alpha)$. In particular, $L^2(\Omega,\alpha,\mathbb{P}|_\alpha)$ is generated by $\mathcal{A}$ as a subspace of $L^2(\Omega, \sigma, \mathbb{P})$.

But what can we say in the general case? Is it possible that the algebra and $\sigma$-algebra generate different subspaces? Relevant references would be appreciated - mostly I'm having trouble searching for literature about actual algebras of random variables because the term '$\sigma$-algebra' appears almost everywhere.

Let $\{v_\lambda:~\lambda\in\Lambda\}$ be a family of real-valued random variables on a probability space $(\Omega, \sigma, \mathbb{P})$. Assume the variables lie in $\bigcap_{p=1}^\infty L^p(\Omega)$ and denote by $\mathcal{A}$ the algebra they generate, and $\alpha$ the sigma-algebra they generate.

If the variables are Gaussian then it is a consequence of the Wiener chaos decomposition (see e.g. Janson - Gaussian Hilbert Spaces) that $\mathcal{A}$ is dense in $L^2(\Omega, \alpha, \mathbb{P}|_\alpha)$. In particular, $L^2(\Omega,\alpha,\mathbb{P}|_\alpha)$ is generated by $\mathcal{A}$ as a subspace of $L^2(\Omega, \sigma, \mathbb{P})$.

But what can we say in the general case? Is it possible that the algebra and $\sigma$-algebra generate different subspaces? Relevant references would be appreciated - mostly I'm having trouble searching for literature about actual algebras of random variables because the term '$\sigma$-algebra' appears almost everywhere.

Let $\{v_\lambda:~\lambda\in\Lambda\}$ be a family of real-valued random variables on a (complete) probability space $(\Omega, \sigma, \mathbb{P})$. Assume the variables lie in $\bigcap_{p=1}^\infty L^p(\Omega)$ and denote by $\mathcal{A}$ the unital algebra generated by them along with the constants, and $\alpha$ the minimal complete sub-sigma-algebra with respect to which they are measurable.

If the variables are Gaussian then it is a consequence of the Wiener chaos decomposition (see e.g. Janson - Gaussian Hilbert Spaces) that $\mathcal{A}$ is dense in $L^2(\Omega, \alpha, \mathbb{P}|_\alpha)$. In particular, $L^2(\Omega,\alpha,\mathbb{P}|_\alpha)$ is generated by $\mathcal{A}$ as a subspace of $L^2(\Omega, \sigma, \mathbb{P})$.

But what can we say in the general case? Is it possible that the algebra and $\sigma$-algebra generate different subspaces? Relevant references would be appreciated - mostly I'm having trouble searching for literature about actual algebras of random variables because the term '$\sigma$-algebra' appears almost everywhere.