Meaning of Exact Transformation and $K$-Automorphism in the context of Ergodic Theory/Mixing

Question

I am reading MAGIC$010$ Ergodic Theory course. In this course's lecture $4$ notes, it is mentioned that
$1)$ Let $T$ be an exact transformation of the probability space $(X,B,μ)$ .Then $T$ is strong-mixing.
$2)$ Let $T$ be a $K$-automorphism of the probability space $(X,B,μ)$. Then $T$ is strong-mixing.

I understand what strong mixing, automorphism and transformation mean but I do not know what "Exact transformation" and "$K$-Automorphism" mean in this context. The author has not defined $K$ anywhere else so I'm not sure if that is a variable. Can someone clear this up? Thanks!

I can attach the link to the notes if required.

Answer 1

These definitions can be found in this article, page 11.

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Definition: A transformation $T$ of $(X, \mathcal B, \mu)$ is exact if $\cap^{\infty}_{n=0}T^{-n}\mathcal B$ consists entirely of null sets and sets of measure $1$.

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Definition: An invertible measure-preserving transformation $T$ of $(X, \mathcal B, \mu)$ is said to be $K$ (for Kolmogorov) if there is a sub-$\sigma$-algebra $\mathcal F$ of $\mathcal B$ such that:

1. $\cap_{n=1}^{\infty}T^{-n}\mathcal F$ is the trivial $\sigma$-algebra up to sets of measure $0$ (i. e. the intersection consists only of null sets and sets of full measure).

2. $\vee_{n=1}^{\infty}T^n\mathcal F = \mathcal B$ (i.e. the smallest $\sigma$-algebra containing $T^n \mathcal F$ for all $n>0$ is $\mathcal B$).

I would not worry that much about these, since they do not appear anywhere else in Charles' lecture notes for Ergodic Theory.