pi-systems and lambda-systems (Dynkin's theorem)
/ DIN-kin /
A constant headache in measure theory is proving that two measures agree everywhere — on every set in a huge sigma-algebra. Checking each set one by one is hopeless. Dynkin's theorem is the slick tool that solves this: it says you only have to verify agreement on a small, easy-to-handle collection of sets (closed under intersection), and the agreement automatically spreads to the whole sigma-algebra those sets generate. It is the workhorse behind uniqueness results all over probability.
The trick splits the two sigma-algebra rules into two weaker structures. A pi-system is a family of sets closed just under finite intersection — for example, all half-lines (-infinity, x], whose pairwise intersections are again half-lines. A lambda-system (or Dynkin system) is a family containing the whole space and closed under complements-within-the-space and under countable disjoint unions — exactly the operations that measure-equality respects, since two measures agreeing on a set agree on its complement, and additivity handles disjoint unions. Dynkin's pi-lambda theorem then states: if a lambda-system contains a pi-system, it contains the entire sigma-algebra generated by that pi-system. Combined with the fact that intersection plus the lambda properties give all the sigma-algebra properties, this is the bridge from easy to hard.
In practice the recipe is: to show two probability measures are equal on the Borel sets, show they agree on all half-lines (-infinity, x] — that is, that their cumulative distribution functions match — note the half-lines form a pi-system, check that the sets where the measures agree form a lambda-system, and Dynkin does the rest. This is exactly why a distribution is completely determined by its CDF, and why independence only needs to be checked on generating events rather than on all events.
To prove two distributions on R are identical, you do not check every Borel set; you only check that P((-infinity, x]) = Q((-infinity, x]) for all x. The half-lines form a pi-system that generates the Borel sets, so Dynkin's theorem upgrades agreement-on-half-lines to agreement everywhere.
Verify on an intersection-closed generating family, and equality propagates to the whole sigma-algebra.
The pi-system must be closed under intersection for this to work; checking agreement on a family that is not intersection-closed does not let you conclude agreement on the generated sigma-algebra.