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My instructor on measure theory solved this problem in class but I am having serious problem in understanding it's proof although I am clear with definitions.

Problem is - Let $P$ be a set and any set $A$ belongs to $P$ iff either cardinality of $A$ is finite or cardinality of Compliment of A is finite. ( it is easy to see $A$ is an algebra) . Define $Q$ to be a set such that any set $B$ belongs to $Q$ iff either Cardinality of $B$ is countable or cardinality of complement of $B$ is countable . ( I proved it to be $ \sigma $ algebra. ) Define $ \sigma ( P)$ = smallest $ \sigma $ algebra containing P.

I have to prove that $ \sigma (P) = Q $.

Can somebody give explanation on how to prove it ?

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Prove that $\mathcal{P} \subset \mathcal{Q}$ and since $\mathcal{Q}$ is a sigma algebra $\sigma({\mathcal{P}})\subset \mathcal{Q}$

Also if $B \in \mathcal{Q}$ and $B$ countable the $B$ is a countable union of finite sets so $B \in \sigma({\mathcal{P}})$

If $B \in \mathcal{Q}$ and $B^c$ is countable then as before $B^c \in \sigma({\mathcal{P}})$ and since $\sigma({\mathcal{P}})$ is a sigma algebra then $B=(B^c)^c \in \sigma({\mathcal{P}})$

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