Difference between a Dynkin System and Sigma algebra

Question

I am currently studying measure theory but there is one thing I simply cannot understand:

The definitions of sigma algebras and dynkin systems are very much alike. However, they differ on the third property:

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So for a Dynkin system, the sets have to be pairwise disjoint to keep the third property correct, whereas for a sigma algebra this is not the case.

However, a dynkin system is said to be a "weaker form" of a sigma algebra. How can this be if they are similar, but for a dynkin system the sets have to be pairwise disjoint and for a sigma algebra this doesn't matter?

I can't just understand this, to me it seems like the pairwise disjoint is just an extra property it has to fulfill.

What am I missing here?

Thanks in advance!

Answer 1

That condition that partly defines a Dynkin system applies to less sets, because you only have to apply this condition to those that are pairwise disjoint. In particular, if you are a $\sigma$-algebra, then you're a Dynkin system.

A toy example - we say we have a $\sigma$-collection of balls if

• every ball costs $5.

we say we have a $\lambda$-collection of balls if

• every ball that weighs a different amount from every other ball, costs $5.

Here, a $\sigma$-collection is always a $\lambda$-collection. But it could be that there are two $30\, \mathrm g$ balls in my $\lambda$-collection, and one costs \$5, and the other costs \$1.

In addition, you can add assumptions to a Dynkin system to turn it into a $\sigma$-algebra. Namely: a Dynkin system $\mathcal D$ that is also closed under finite intersections is a $\sigma$-algebra. That is - suppose $A_i$ is a countable collection of sets. Define $B_1 = A_1$ and inductively $$B_i = A_i \cap A_{i-1}^c \cap \dots \cap A_1^c \in \mathcal D.$$ But for $i<j$, $$B_i \cap B_j \subset A_i \cap B_j = \emptyset$$ so now by the Dynkin property, $$ \bigcup_{i=1}^\infty A_i = \bigcup_{i=1}^\infty B_i \in \mathcal D $$ so $\mathcal D$ is closed under countable unions.

See also : (a) this example that shows that there exist Dynkin systems that are not $\sigma$-algebras, and (b) the $\pi-\lambda$ theorem of Dynkin.