I was wondering why Borel algebra $B(X)$ is defined to be the smallest sigma algebra containing all open subsets of X.
If it contains all the subsets, then how can any other sigma algebra have more subsets? I'm asking why the word 'smallest' is used in definition.
[Math] Borel sigma algebra – why smallest
measure-theoryprobability
Best Answer
It doesn't have to include all the subsets of $X$. We can prove, in the case of $\Bbb R$ for example, that the Borel sets is a collection whose cardinality is the same as $\Bbb R$. Since Cantor's theorem tells us that there are many more subsets to $\Bbb R$ it follows that most subsets are not Borel sets.
To see a much more degenerate example, consider any space with the trivial topology. It's Borel subsets are the open sets, $\varnothing$ and $X$ itself.