Generated Sigma-Algebras Need Not Converge in Mean Square under Convergence of the Generating Random Variables

There exist a probability space $(\Omega,\mathcal{F},P)$, random variables $U$ and $V$ on it, and a sequence $(U^n)_{n\in\mathbb{N}}$ of random variables on it with the following properties, where $\sigma(\cdot)$ denotes the $\sigma$-algebra generated by a random variable, measurability of a random variable with respect to a sub-$\sigma$-algebra is as in Existence and Uniqueness of Conditional Expectation for Square-Integrable Random Variables, and $\lVert\cdot\rVert_{2}$ is the mean-square norm:

(a) (Strong convergence of the generating variables) $U^n(\omega)\to U(\omega)$ for every $\omega\in\Omega$, and the real sequence $(\lVert U^n-U\rVert_{2})_{n\in\mathbb{N}}$ has limit $0$; all of $U$, $V$, $U^n$ are square-integrable.

(b) (Adaptedness before the limit) For every $n\in\mathbb{N}$, the random variable $V$ is $\sigma(U^n)$-measurable.

(c) (Failure of adaptedness in the limit) $V$ is not $\sigma(U)$-measurable. In particular, the constant sequence $X^n=V$ consists of $\sigma(U^n)$-measurable square-integrable random variables and converges in mean square to $V$, yet the limit is not measurable with respect to $\sigma(U)$, the $\sigma$-algebra generated by the limit of the generating variables.

(d) (Failure of mean-square convergence of the $\sigma$-algebras) $\sigma(U^n)$ does not converge in mean square to $\sigma(U)$. Quantitatively, in the notation of Conditional Expectation of a Square-Integrable Random Variable: every conditional expectation of $V$ given $\sigma(U^n)$ equals $V$ almost surely, every conditional expectation of $V$ given $\sigma(U)$ equals the constant $\tfrac12$ almost surely, and

Consequently, convergence of random variables $U^n\to U$, even simultaneously pointwise on all of $\Omega$ and in mean square, does not imply that mean-square limits of $\sigma(U^n)$-measurable random variables are $\sigma(U)$-measurable, nor that conditional expectations given $\sigma(U^n)$ converge to conditional expectations given $\sigma(U)$.