what is difference between Borel algebra and Borel σ-algebra?

Question

I am a university student, professor gave us definitions of Borel algebra and Borel σ-algebra. In Wikipedia they are the same see here.

but professor defined Borel algebra as: let $\Omega$ be $[a,b)$ set on real line. Borel algebra on $\Omega$ is finite intersection of such $[c1,c2), [c1,c2], (c1,c2], (c1,c2)$ intervals.(in wikipedia it is not finite it is countable, please help me which one is correct). are in fact Borel algebra and Borel σ-algebra same? please give me real definitions.

Answer 1

Algebras are closed under finite unions and intersections; $\sigma$-algebras are closed under countable unions and intersections.

The other requirements are that algebras include $\Omega$ and $\emptyset$, and are closed under complementation.

Borel algebras are a particular example, i.e. with the half open intervals serving as a subbase.

Answer 2

For instance consider $\Omega = \mathbb{R}$. Let $\mathcal{I}$ be the collection of all half open intervals $]a,b]$ with $a < b$. The $\sigma$-algebra generated by $\mathcal{I}$ is the smallest $\sigma$-algebra containing all sets in $\mathcal{I}$. Such a $\sigma$-algebra always exists. That one is called the Borel $\sigma$-algebra.

It contains at least all other types of intervals because

$$]a,b[ = \displaystyle\bigcup\limits_{m\geq k}\left(\left]a,b-\dfrac{1}{m}\right]\right)$$

as a countable union. Hence it also contains all sets

$$[a,b[ = \displaystyle\bigcap\limits_{m\in\mathbb{N}_0}\left(\left]a-\dfrac{1}{m},b\right[\right)$$

and also all sets

$$[a,b] = \displaystyle\bigcap\limits_{m\in\mathbb{N}_0}\left(\left[a,b+\dfrac{1}{m}\right[\right)$$

as countable intersections.

Hope this helps,