# Borel subset of $\mathbb{R}$ (terminology)

I am confused with the following terminologies in the context of probability and measure:

1. Borel subset of $$\mathbb{R}$$.

2. Borel set on reals.

3. Borel $$\sigma$$-algebra of subsets of $$\mathbb{R}$$.

How are they different, and how do we use this (w/examples)?

The way I understand construction of the Borel $$\sigma$$-algebra on [0,1] is:

Start with [0,1] and a closed interval $$[a,b]\in[0,1]$$. Add everything else by looking at the union of a sequence of closed intervals, and the like IN ORDER to have a sigma algebra. This sigma algebra is called the Borel sigma algebra of subsets of $$[0,1]$$. The sets in this sigma algebra are called Borel sets.

In the context of probability, every subset of [0,1] is a Borel set, so the probability is determined on these subsets.

So, does $$B[0,1]$$ look something like, $$\{[0,1],\emptyset, \{,,\},\{,,,\},\bigcap_n[,]_n,\bigcup_m(,)_m,...\}$$, and each element in $$B[0,1]$$ is called Borel set? Then what is the Borel subset of $$[0,1]$$?

• It is not really possible to write a generic Borel subset, that is why we describe the Borel $\sigma$-algebra as the $\sigma$-algebra generated by e.g. closed intervals $[a,b]$. Nov 15, 2019 at 0:47
• @Math1000 I am mainly asking regarding the terminologies here though Nov 15, 2019 at 0:52
• I think 1=2 and are elements of 3. PS your description is a little imprecise. There are other sigma algebras that contain all the closed intervals. If you add literally every subset of $[0,1]$, obtaining the powerset, you still have a sigma algebra. The Borel sigma algebra is the smallest sigma algebra (ordered by inclusion of sigma algebras) that contains closed intervals $[a,b].$ Similarly its not true at all that every subset of $[0,1]$ is a Borel set Nov 15, 2019 at 0:56

The Borel $$\sigma$$-algebra on $$\mathbb R$$ (your point 3) is a collection of sets. Each one of the sets in this $$\sigma$$-algebra is called a Borel set (both 1 and 2 are equivalent ways to call a Borel set).
As said it is hard to describe explicitly the structure of a Borel set. It is more convenient to stick with the more abstract definition "the smallest $$\sigma$$-algebra which contains all the intervals $$[a,b]$$".
I suggest you to have a look at the following lecture, I find it very clear. $$\sigma$$-algebras are at page 4 (note that $$\sigma$$-field is a synonym of $$\sigma$$-algebra). The Borel $$\sigma$$-algebra is at the bottom of page 6.