I am currently reading the book "Probability Theory:A Comprehensive Course" by Achim Klenke,in which there is a theorem left as an exercise for the reader.
Definition 1:
Let $\Omega$ be a nonempty set,and let $\mathcal A$ be a collection of subsets of $\Omega$.Then $\mathcal A$ is a Dynkin system if
(i) $\Omega \in \mathcal A$,
(ii) $A ^c \in \mathcal A$ if $A \in \mathcal A$,
(iii) If $A_1,A_2,A_3,…$ is a sequence of subsets in $\mathcal A$ such that $A_i \cap A_j=\emptyset$ for all $i \neq j$,then $\bigcup_{n=1}^\infty \in \mathcal A$.
Definition 2 :
Let $\mathcal A \subset 2^\Omega$ be an arbitrary class of subsets of $\Omega$ and let $A \in 2^\Omega \backslash \{ \emptyset \}$
.The class
$\mathcal A \vert _A := \{ A \cap B:B \in \mathcal A \} \subset 2^A$
is called the trace of $\mathcal A$ on $A$.
Theorem:
Let $A \subset \Omega$ be nonempty and let $\mathcal A$ be a Dynkin system on $\Omega$,then $\mathcal A \vert _A$ is not necessarily a Dynkin system on $A$ unless $A=\Omega$.
My Question:
The theorem stated above is the second half of the desired theorem from the book,and I failed in finding a counterexample that violates the third axiom of the so-called Dynkin system to complete the proof.
Can someone show me a counterexample? Thank you!
Best Answer
$\Omega = \{1,2,3,4\}$, $\mathcal A=\{\varnothing, \{1,3\},\{2,4\},\{1,4\},\{2,3\},\Omega\}$, $A = \{1,2,3\}$.