A noise is a kind of homomorphism from a Boolean algebra of domains to the lattice of $\sigma$-fields. Leaving aside the homomorphism we examine its image, a Boolean algebra of $\sigma$-fields. The largest extension of such Boolean algebra of $\sigma$-fields, being well-defined always, is a complete Boolean algebra if and only if the noise is classical, which answers an old question of J. Feldman.
"Noise as a Boolean algebra of $\sigma$-fields." Ann. Probab. 42 (1) 311 - 353, January 2014. https://doi.org/10.1214/13-AOP861