3

Question: How do I show that a $\sigma$-algebra is either finite or uncountable?

I am given the hint that $\mathcal{P}(\mathbb{N})$ is uncountable.

What I think I should do:

Suppose that S is a set and let $\mathcal{A}$ be a $\sigma$-algebra on S. We know that $\mathcal{A}\subseteq \mathcal{P}(S)$, where $\mathcal{P}(S)$ denotes the power set of $S$. I think I need to distinguish a couple of cases:

  • The case where $\mathcal{P}(S)$ is uncountable and thus infinite.

  • The case where $\mathcal{P}(S)$ is countable and infinite.

  • The case where $\mathcal{P}(S)$ is countable and finite.

I feel like my next step would be to show what each case would imply for $\mathcal{A}$, but I don't really know how.

titusAdam
  • 2,847
  • $\mathcal{P}(S)$ can not be countable and infinite. The "smallest" infinite set is $\mathbb{N}$ and $\mathcal{P}(\mathbb{N})$ is already uncountable. As you know from the hint. Or Cantor's lemma. – Cornman Nov 23 '17 at 12:54
  • You need just two cases. What if $S$ is finite. How many elements can $\mathcal{A}$ have at most? What if $S$ is infinite? – Cornman Nov 23 '17 at 12:55
  • 1
    Try showing that if $\mathcal{A}$ is infinite then it must contain an infinite sequence of disjoint sets. From this, you can conclude that your $\sigma$-algebra is uncountable more easily. – Karina Livramento Nov 23 '17 at 13:17

0 Answers0