Measure Preserving Transformation Induced by a $*$-automorphism on $L^\infty(X,\mu)$

Question

The following excerpt is from Connes' Noncommutative Geometry

Let $(X, \mathcal{B}, \mu)$ be a standard Borel space equipped with a probability measure $\mu$, and let $\ T$ be a Borel transformation of $(X, \mathcal{B})$ that leaves $\mu$ invariant. Let $M = L^\infty(X, \mathcal{B}, \mu)$ and let $\varphi$ be the state associated with $\mu$. Then $T$ determines an automorphism of $M$ that preserves $\varphi$, by the equality $$\theta(f) = f \circ T^{−1}.$$ Conversely, every automorphism of $M$ that preserves $\varphi$ can be obtained in this way.

My question concerns the "conversely" part of the above excerpt: How does one obtain a measure-preserving transformation (henceforth m.p.t) $T$, given the automorphism $\theta$ on $M$?

One route that I've taken is to identify $X$ with $\hat{X}$, the character space of $L^\infty$. In other words, $\hat{X}:=\{\hat{x}:L^\infty(X,μ)\rightarrow \mathbb{C}\mid \hat{x}(f)=f(x)\}$ consists of point evaluations. The character space $\hat{X}$ is a compact Hausdorff space under the weak$^*$ topology inherited from the dual space of $L^\infty$. Define $T:\hat{X}\rightarrow \hat{X}$ by the prescription $T(\hat{x})=\hat{x}\circ \theta^{-1}$. I think this is the m.p.t that I am looking for. However, I do not know how to prove that T is indeed an m.p.t.

If such a result is true, then it must be very well-known. A reference to the proof of such a result would be gladly appreciated.

EDIT: In light of Blackburne's answer I've added the "descriptive-set-theory" tag.

Answer 1

1. Every idempotent on $L^\infty$ is multiplication by (the equivalence class of) the characteristic function of a measurable set

2. Such an automorphism maps an idempotent onto another idempotent.

3. This gives your result at the level of sets, i.e., it provides an isomorphism of the Boolean $\sigma$-algebra of equivalence classes of measurable sets.

4. This works in a more general situation than the one described in your query.

5. If you wish to get the result at the pointwise level, then you indeed require the fact that you have a standard measure space. At this level, the result is purely measure-theoretical and the details can be found in the text by Kechris ("Classical Descriptive Set Theory").

Edit after comment. At the basis of the the above dichotomy is a "Richtungstreit" in the foundations of probability as to whether one should regard points or events as the basic entities, more precisely is a measure defined on a $\sigma$-algebra of subsets of a set $\Omega$ or on a Boolean $\sigma$ algebra, specifically the algebra of equivalence classes of such subsets? Much can be said for the latter approach and it has been lucidly presented by, for example, David Fremlin in his "Topological Riesz spaces and Measure Theory". I don't have access to Kechris at the moment but the result which establishes the relationship between the two approaches for standard measure spaces is apparently in section 17.F. Another possible reference is Petersen, "Ergodic Theory", Th. 4.7.