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I will refer to the three definitions of approximate continuity given as AC0, AC1 and AC2 respectively and show they are all equivalent.
AC1 iff AC2: Given a set $A$ by AC1, $A$ has density 1, $A$ is ...

Le Gall's "Brownian Motion, Martingales and Stochastic Calculus" may be a good fit. You need to be comfortable with functional analysis though, cause he uses it sometimes to streamline proofs.

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