Dynkin System/Semirings/Rings: Why? What good for? 


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*Why do we need Dynkin Systems? Sure they're per se $\sigma$-algebras whenever they're intersection stable. But why considering them at all?
Is there, maybe, any existence theorem for measures based on Dynkin Systems? Or is there sth that tells us that we need to consider them?

*Moreover, is there any other reason apart from historical ones for considering Semirings? So far, the Kolmogorov Extension Theorem for premeasures seems reasonable. However, is there other deep arguments for the need of semirings?

*What about Rings themselves?
Again, I want to stress that, I'm not considering historical arguments but rather the ones that say we have to or we need to or at least we should consider Dynkin Systems, Semiring and Rings rather than merely $\sigma$-algebras.
 A: Let $(X,\mathcal{X})$ and $(Y,\mathcal{Y})$ be measurable spaces. A measurable rectangle is a subset of $X\times Y$ of the form $A\times B$ with $A\in\mathcal{X}$ and $\mathcal{Y}$. One usually endows $X\times Y$ with the product $\sigma$-algebra, which is the $\sigma$-algebra generated by measurable rectangles. If we have finite measures $\mu$ and $\nu$ on $X$ and $Y$ respectively, the product measure $\mu\otimes\nu$ on the product $\sigma$-algebra is given by $\mu\otimes\nu(A\times B)=\mu(A)\nu(B)$ and then extended. The measurable rectangles, however, form only a semi-ring.
Dynkin systems are mainly useful because of the $\pi-\lambda$ Theorem. This result is very useful to show that two measures coincide. For example, any two Borel probability measures on the real line coincide if they agree on intervals. This allows us to define probability distributions by cumulative distribution functions.
A: I remember an exercise for which it was very essential to use a Dynkin system:
Show that $\mathcal{D}$ is a Dynkin-System (concerning transition measure)
So it might appear like a rather theoretical definition but it has indeed practical use.
