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Let $X$ be a finite set. Let $\Sigma \subseteq P(X)$ be a finite sigma algebra on $X$. I have been able to prove that the cardinality of $\Sigma$ is always an even number: the collection $$ \left\{\left\{Y{,}\ X\setminus Y\right\}:Y\in \Sigma\right\}$$ is a partition of $\Sigma$. Thus, $$\left|\Sigma\right|=\left|\left\{\left\{Y{,}\ X\setminus Y\right\}:Y\in \Sigma\right\}\right|\cdot\left|\left\{Y{,}\ X\setminus Y\right\}\right|=2\left|\left\{\left\{Y{,}\ X\setminus Y\right\}:Y\in \Sigma\right\}\right|{,}$$ and the cardinality of $\Sigma$ is necessarily an even number.

However, there is more: the cardinality of $\Sigma$ is a power of $2$. This I am unable to prove. What would be an elementary way of showing that fact? I guess that one should construct some collection so that $\Sigma$ is its power set, thus having a cardinality of some power of $2$.

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Hints: For each $x \in X$ there is a smallest member $A_x$ of the sigma algebra containing $x$. $A_x$ and $A_y$ are either equal or disjoint. Any set in the sigma algebra is a union of $A_x$'s. This gives a one-to-one function from the sigma algebra onto a power set.

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Define on $X$ the equivalence relation $x \sim y$ if and only if for each $A\in\Sigma$, $\mathbf{1}_A(x)=\mathbf{1}_A(y)$. For any $x\in X$, the equivalence class of $x$ is $C_x:=\bigcap_{A\in \Sigma\,:\,x\in A}A$ and since $\Sigma$ is finite, we know that $C_x$ belongs to $\Sigma$. The collection of equivalence classes of elements of $X$ forms a finite partition, says $A_i, 1 \leqslant i\leqslant N$, of elements of $\Sigma$.

For $I\subset\{1,\dots,N\}$, let $B_I=\bigcup_{i\in I}A_i$ if $I$ is not empty and $B_\emptyset=\emptyset$. Then the map $\tau\colon \mathcal P(\{1,\dots,N\})\mapsto \Sigma$ given by $\tau(I)=B_I$ is a bijection.

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