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One of the questions on a past final for a measure theory course I'm taking is "Let X be a set. How many $\sigma$-algebras of subsets of X contain exactly $5$ elements?". Would the answer to this question just be $infinity$ since we don't have any information about the set X? Apologies if this question may seem obvious but I find measure theory extremely difficult to wrap my head around.

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    $\begingroup$ Can you come up with an example of a set $X$ and a $\sigma$-algebra on $X$ that has 5 elements? $\endgroup$ – Alonso Delfín Jul 31 '20 at 22:57
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    $\begingroup$ By any chance is it impossible to have a $\sigma$-algebra on X with 5 elements since we must always have a set and its compliment in that $\sigma$-algebra which would thus mean there is always an even number of elements? If I'm way off I apologize! $\endgroup$ – Jack Jul 31 '20 at 23:03
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    $\begingroup$ Yes! In general, it can be shown that any $\sigma$-algebra with a finite number of elements must have $2^n$ elements for some $n$. If you are interested, I can add a sketch of a proof for this fact as an answer. $\endgroup$ – Alonso Delfín Jul 31 '20 at 23:05
  • $\begingroup$ @AlonsoDelfín There is one, here: math.stackexchange.com/questions/181555/… $\endgroup$ – uniquesolution Jul 31 '20 at 23:07
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    $\begingroup$ Thank you so much for your help! A sketch would be amazing, thank you. $\endgroup$ – Jack Jul 31 '20 at 23:07
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The answer is none.

The reason is the following theorem:

Theorem. If $\mathcal{F}$ is a $\sigma$-algebra on $X$ that has a finite number of elements, then $\mathrm{card}(X)=2^n$ for some $n$.

I'll give a sketch of a proof and I'll leave it to you to fill in all the missing details.

Sketch of proof. Assume w.l.o.g. that $X \neq \varnothing$. For each $x \in X$ let $$ \mathcal{F}_x:=\{ E \in \mathcal{F} : x \in E\} $$ Clearly each $\mathcal{F}_x \neq \varnothing$. We say $x \sim y$ in $X$ if $\mathcal{F}_x = \mathcal{F}_y$. This gives an equivalence relation on $X$. Some useful facts that you can check are:

  • that the equivalence classes are actually given by $$ [x]=\bigcap_{E \in \mathcal{F}_x}E \in \mathcal{F} $$
  • that for each $E \in \mathcal{F}$ $$ E=\bigsqcup_{x \in E} [x] $$ where $\bigsqcup_{x \in E}$ means that the union is disjoint (i.e. if $x, y \in E$ are such that $x \neq y$, then $[x] \neq [y]$).

Let $\mathscr{A}$ be the set whose elements are distinct equivalence classes of $\sim$ and put $n := \mathrm{card}(\mathscr{A})$. If you want a clear picture, this means that after selecting $n$ representatives you have $$ \mathscr{A}=\{[x_1], \ldots, [x_n]\} $$ Let $\mathcal{P}(\mathscr{A})$ be the power set of $\mathscr{A}$. From basic set theory we know that $\mathrm{card}(\mathcal{P}(\mathscr{A}))=2^n$. Finally, define a map $\varphi: \mathcal{P}(\mathscr{A}) \to \mathcal{F}$ by letting $$ \varphi(\mathscr{B}):= \bigsqcup_{[x] \in \mathscr{B}} [x] $$ for any $\mathscr{B} \subseteq \mathscr{A}$ with $\mathscr{B}\neq \varnothing$ and $\varphi(\varnothing):=\varnothing$. It is not hard at this point to check that $\varphi$ is a bijection and therefore that $\mathrm{card}(\mathcal{F})=2^n$. $\blacksquare$

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