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This is essentially Exercise 1.3.1 from Durrett's probability theory textbook. In fact strict equality is true, but the other direction of inclusion is so trivial that I decided to de-emphasise it in the question.

The biggest problem for me is that $\sigma(\mathcal A)$ looks quite intractable, and I need a more explicit representation. I'd like to guess that every set in $\sigma(\mathcal A)$ can be written as the output of countably many $A_n\in \mathcal A$ under countably often union, intersection and complements. But I lack the proof, or even the rigorous formulation of such an idea. Instead of attempting to characterise $\sigma(\mathcal A)$ which might be too large to, is there any smarter way to solve the problem?

Vim
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    This follows from properties of the preimage, and some set theory. A full answer can be found here http://math.stackexchange.com/questions/7881/preimage-of-generated-sigma-algebra – Andres Mejia Nov 16 '16 at 14:23
  • Proposition 1.4.1 in the following text should also do it, the point is that the preimage operation commutes with unions and intersections http://www.uio.no/studier/emner/matnat/math/MAT2400/v13/mathanalbook.pdf – Andres Mejia Nov 16 '16 at 14:25
  • @AndresMejia yeah I understand that $f^{-1}$ preserves union, intersection and complements. Thanks for the resources. – Vim Nov 16 '16 at 14:30

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