Dynkin System Closed under Set Difference

Source Work

 * $5$: Uniqueness of measures
 * Problem $5.3$
 * Problem $5.3$

This mistake can be seen in the first edition (2005), reprinted in 2007: ISBN 0-521-85015-5 (hardback) and ISBN 0-521-61525-9 (paperback).

Mistake

 * "Let $\mathcal D$ be a Dynkin system. Show that for all $A, B \in \mathcal D$ the difference $B \setminus A \in D$."

This is, however, in general, only true if it is also stipulated that $A \subseteq B$, which case is proved on:
 * Dynkin System Closed under Set Difference with Subset

The falsehood of the general statement is demonstrated by the following example:

Let $X = \left\{{1, 2, 3, 4}\right\}$.

Define $\mathcal D$ to be the collection of subsets of $X$ having an even number of elements:


 * $\mathcal D := \left\{{S \subseteq X: \text{$\# \left({S}\right)$ is even}}\right\}$

As for example $\left\{{1, 2}\right\}$ and $\left\{{2, 3}\right\}$ are in $\mathcal D$, it is seen that $\mathcal D$ is not closed under set difference.

It remains to show that $\mathcal D$ constitutes a Dynkin system:


 * It is clear that $X \in \mathcal D$ as $\# \left({X}\right) = 4$.
 * Suppose that $\# \left({S}\right) = 2k$ for $k \in \N$. Then $\# \left({X \setminus S}\right) = 4 - 2k$, which is also even. Hence $X \setminus S \in \mathcal D$.
 * Under disjoint union, the cardinalities are added, and Sum of Even Integers is Even guarantees that the disjoint union is again in $\mathcal D$.

Hence $\mathcal D$ is a Dynkin system, and it provides a counterexample to the claim.

Acknowledgements
This error has been corrected in the third (2010) and later printings.

It appears in the online list of misprints maintained by Schilling.