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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 $[0,1]$?

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  • $\begingroup$ 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 $[a,b]$. $\endgroup$
    – Math1000
    Nov 15, 2019 at 0:47
  • $\begingroup$ @Math1000 I am mainly asking regarding the terminologies here though $\endgroup$ Nov 15, 2019 at 0:52
  • $\begingroup$ 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 $\endgroup$ Nov 15, 2019 at 0:56

1 Answer 1

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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.

https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-436j-fundamentals-of-probability-fall-2018/lecture-notes/MIT6_436JF18_lec01.pdf

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  • $\begingroup$ good reference. thanks!! $\endgroup$ Nov 15, 2019 at 20:38

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