Yet another question about the definition of the Borel $\sigma$-field on $\Bbb R$

Question

I am reading Capinski and Kopp's Measure, Integral and Probability. They present the following definition for the Borel $\sigma$-field $$ \mathcal B=\bigcap\{\mathcal F:\mathcal F\text{ is a $\sigma$-field containing all intervals}\}. $$ So I believe I understand correctly that $\mathcal B$ is the intersection of a family of $\sigma$-fields which must also be a sigma field. What I don't understand is how this family, i.e. $\{\mathcal F:\mathcal F\text{ is a $\sigma$-field containing all intervals}\}$ can containg more than one $\sigma$-field. How can there be more than one $\mathcal F$ with the property that it contains all intervals in $\Bbb R$?

Answer 1

There are many sigma fields containing all intervals. The power set of $\mathbb R$ is one. Lebesgue measurable sets is another and the Borel sigma field is yet another. These three are distinct.