Definition:
The Borel $\sigma$-algebra on $\mathbb R$ is the $\sigma$-algebra B($\mathbb R$) generated by the $\pi$-system $\mathcal J$ of intervals $\ (a, b]$, where $\ a<b$ in $\mathbb R$ (We also allow the possibility that $\ a=-\infty\ or \ b=\infty$) Its elements are called Borel sets. For A $\in$ B($\mathbb R$), the $\sigma$-algebra$$B(A)= \{B \subseteq A: B \in B(\mathbb R)\}$$of Borel subsets of A is termed the Borel $\sigma$-algebra on A.
I struggle with this part especially "generated by the $\pi$-system $\mathcal J$ of intervals (a, b]"
In addition could someone please provide an example of a Borel set, preferably some numerical interval :)
Also is $\mathbb R$ the type of numbers that the $\sigma$-algebra is acting on?
$\endgroup$2 Answers
$\begingroup$Ignore the phrase "$\pi$-system" for the time being : What you are given is a collection $\mathcal{J}$ of subsets of $\mathbb{R}$ and the $\sigma$-algebra you seek is the smallest $\sigma$-algebra that contains $\mathcal{J}$. This is the definition of the Borel $\sigma$-algebra. For example $\{1\}$ is a Borel set since $$ \{1\} = \bigcap_{n=1}^{\infty} (1-1/n,1] = \mathbb{R}\setminus \left(\bigcup_{n=1}^{\infty} \mathbb{R}\setminus (1-1/n,1]\right) $$ Does this help you understand what this $\sigma$-algebra can contain? It is not possible to list down all the elements in $B(\mathbb{R})$ though.
Now, the reason we choose this $\sigma$-algebra is simple : We want continuous functions to be measurable - a rather reasonably requirement which is often imposed when dealing with measure spaces that are also topological spaces.
$\endgroup$ 2 $\begingroup$Borel $\sigma$-algebras turn op if we are working on topological spaces. If $X$ denotes a topological space and $\tau$ is its topology then the smallest $\sigma$-algebra that contains $\tau$ (in other words: the $\sigma$-algebra generated by $\tau$) is the Borel $\sigma$-algebra on that space. In special case $X=\mathbb R$ equipped with its usual topology it can be shown that the $\sigma$-algebra generated by the open sets (by definition the Borel $\sigma$-algebra) coincides with the $\sigma$-algebra generated by intervals $(a,b]$.
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