TITLE: Why pi-systems and Dynkin/lambda systems? On the relative merits of approaches in measure theory.
QUESTION [21 upvotes]: What is the point of $\pi$-systems and
  $\mathcal{D}$ / Dynkin /
  $\lambda$-systems? 

I am an analyst in the process of consolidating my measure theory knowledge before moving on to harder/newer things, having been first introduced to measure theory in a course with a probability as opposed to an analysis viewpoint. So far, everything that I've needed from elementary measure theory for analysis can be done (and is done in all of my analysis textbooks) without mention of the $\pi$-systems and $\mathcal{D}$-systems which were used in my first course. Do these set systems belong strictly to probability and not analysis? Heuristically, are they useful or important in any way? Why?

REPLY [4 votes]: I think there's indeed a lot of confusion here, in that in many situations the $\pi-\lambda$ is applied where a weaker result would suffice. What one uses most of the time is not the $\pi-\lambda$ theorem itself, but the following corollary: 

If $\mu_1$ and $\mu_2$ are two probability  measures that agree on a
  $\pi$-system, then they agree on the $\sigma$-algebra generated by
  that $\pi$-system.

The example about independent events given above by Peter Luthy falls into this category.
In many textbook situations the $\pi$-system in question is actually a semi-ring. For ($\sigma$-)finite premeasures, Carathéodory extension is unique for very simple reasons: any extension of the measure is bounded from above by the outer measure, and by passing to the complement it is also bounded from below. Thus, for semi-rings the above corollary is trivial.
A typical example of a $\pi$-system that is not a semi-ring are closed sets in a topological space. But here, the fact that a Borel measure is determined by its values on closed sets follows from regularity, which does not require $\pi-\lambda$ theorem either.
One place where you need $\pi-\lambda$ theorem (or a similar result) in an essential way is Fubini. There, you have a $\pi$-system (which is also a semi-ring) of sets of the form $E_1\times E_2$, and two ways to extend the pre-measure: by product-measure (= Carathéodory extension), and by integrating the measures of the slices. What is tricky to show is that the collection of sets for which the latter is well defined is a $\sigma$-algebra, in particular, that it is closed under pairwise intersections. Indeed, it's not clear under such operation, the function under the integral remains measurable. The $\pi-\lambda$ theorem allows one to bypass this difficulty in a very neat way.
That said, once you have the $\pi-\lambda$ theorem at hand, it is often just shorter to write down the proof that something is a $\lambda$-system than that it is a $\sigma$-algebra. No wonder the authors of textbooks use it systematically even when it's an overkill.