Dynkin System Closed under Set Difference

Source Work

$5$: Uniqueness of measures

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).

Let $\DD$ be a Dynkin system. Show that for all $A, B \in \DD$ 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:

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

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

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

$\DD := \set {S \subseteq X: \text {$\map {\#} S$ is even}}$

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

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

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

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

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

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