I was reading


and there's this quote:

A measurable function $f : \Omega \to \mathbb{R}$ is called generating if $\mathcal{F}$ is the completion of the σ-algebra of inverse images $f^{-1}(B)$, where $B \subset \mathbb{R}$ runs over all Borel sets.

Caution. The following condition is not sufficient for $f$ to be generating: for every $A \in \mathcal{F}$ there exists a Borel set $B \subset \mathbb{R}$ such that $P ( A \Delta f^{-1}(B) ) = 0$.

I don't get the second paragraph. The condition in the second paragraph looks like an equivalent condition to me.


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