TITLE: What is an example of a lambda-system that is not a sigma algebra?
QUESTION [30 upvotes]: What is an example of a lambda-system that is not a sigma algebra?

REPLY [6 votes]: For another example, let $(\Omega, \mathcal{F}, P)$ be a probability space and fix some event $A \in \mathcal{F}$.  Let $\mathcal{L}$ be the collection of all events which are independent of $A$, i.e.
$$\mathcal{L} = \{ B \in \mathcal{F} : P(A \cap B) = P(A) P(B)\}.$$
It is not hard to check that $\mathcal{L}$ is a $\lambda$-system.  To see it need not be a $\sigma$-algebra, take as in my other answer a probability space $\Omega = \{HH, HT, TH, TT\}$, $\mathcal{F} = 2^\Omega$, $P(A) = \frac{1}{4} |A|$ consisting of two independent fair coin flips.  Set $A = \{HH, HT\}$, the event that the first coin is heads.  Then $\{HH, TH\}, \{HH, TT\}$ are in $\mathcal{L}$ but their union $\{HH, TH, TT\}$ is not.
Incidentally, this is really of the same form as my other answer if we take $Q$ to be the conditional probability measure $Q(E) = P (E \mid A) = P(E \cap A)/P(A)$.  (Except when $P(A)=0$, but that case is trivial.)