See section 1.4 of enter link description here. Here is an excerpt from that section:
...two probability measure spaces $(X_1, \mu_1)$ and $(X_2,\mu_2)$ are isomorphic if there exist conull subsets $Y_1$ and $Y_2$ of $X_1$ and $X_2$, respectively, and a Borel isomorphism $\theta : Y_1 \to Y_2$ such that $\theta_{*} \mu_1 |_{Y_1} = \mu_2 |_{Y_2}$, i.e., $(\theta_{*} \mu_1 |_{Y_1})(E) = \mu_1(\theta^{-1}(E)) = \mu_2(E)$ for every Borel subset $E$ of $Y_2$. Such a map $\theta$ is called a probability measure preserving (p.m.p.) isomorphism, and a p.m.p. automorphism whenever $(X_1, \mu_1) = (X_2,\mu_2)$...We denote $\text{Aut}(X,\mu)$ the group of (classes modulo null sets of ) p.m.p. automorphisms of a probability measure space $(X,\mu)$.
First, what exactly is a Borel isomorphism? Next, I'm guessing that $(\theta_{*} \mu_1 |_{Y_1})(E) = \mu_1(\theta^{-1}(E))$ is a definition. Is that right? Finally, I having trouble making sense of the definition of $\text{Aut}(X,\mu)$. Presumably, the group operation is function composition, but how does composition make sense when these p.m.p. automorphisms are defined on subsets of $X$ (conull subsets to be exact)? The domains ranges might not be compatible, no?
A Borel isomorphism is a measurable bijection with measurable inverse. (Usually this term is only used for standard Borel spaces, I'm not sure if there is such an assumption here).
Yes, that's the definition of pushforward measure.
Elements of $\mathrm{Aut}(X,\mu)$ are equivalence classes of functions that agree modulo a null set. If $f$ is defined on $Y_1$ and $g$ is defined on $Y_2$ you can take $g\circ f$ to be defined on $f(Y_1)\cap Y_2$ (this makes sense because $f(Y_1)$ is still full measure).