Let P be the statement: Every subset of plane belongs to the sigma algebra generated by $\{A \times B : A, B \subseteq \mathbb{R}\}$.

Let Q be the statement: Every continuum-sized family of subsets of $\mathbb{R}$ is contained in a countably generated sigma-algebra.

Both statements are independent of ZFC and P implies Q. Does Q imply P?

This stems from the question [Countably generated sigma algebras][1]. As noted by Gro-Tsen, it was also asked in  [Bing, Bledsoe & Mauldin, "Sets Generated by Rectangles", Pacific J. Math. 51 (1974) 27–36](https://projecteuclid.org/euclid.pjm/1102912790) ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=357124)).

Arnie Miller has a related result: Letting $\kappa = \aleph_{\omega_1}$, it is consistent that $\mathfrak{c} = \kappa^+$, every $\kappa$-sized family of sets of reals is contained in a countably generated sigma algebra and there is a subset of $\kappa \times \mathfrak{c}$ which is not in the sigma-algebra of abstract rectangles. See Theorem 5.12 in the notes [Miller - Borel hierarchies][2] or Theorem 55 in his thesis [Miller - On the lengths of Borel hierarchies][3] ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=548475)).


  [1]: https://math.stackexchange.com/questions/2081480/countably-generated-sigma-algebras
  [2]: http://www.math.wisc.edu/~miller/res/bor.pdf
  [3]: http://www.math.wisc.edu/~miller/res/hier.pdf