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

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.

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.

• Normally when you say Borel algebra you actually mean the Borel $\sigma$-algebra. Do you have examples from the literature of anyone using the Borel algebra in a different meaning? Nov 4, 2020 at 18:40
• thank you, great explanation, and can you add some more information? such as Borel σ-algebra and little bit about Borel algebra. Nov 4, 2020 at 18:40

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,

• what about borel algebra? @Werner Nov 4, 2020 at 19:05
• Never encountered it. Typically in the Borel case a sigma-algebra is being taken. Nov 5, 2020 at 20:04