What is, in layman terms, a Borel $\sigma$-algebra? I am aware question is highly correlated with What is the significance of a Borel $\sigma$-algebra?
However, I am looking for a high-level explanation about Borel $\sigma$-algebra. Probabilities can be explained very intuitively to an average person, I am wondering whether also this concept can be demystified without using too much mathematical jargon.

A Borel subsets of $\mathbb R$ is any subset that can be obtained by starting with open sets and applying the operations of union of finite or infinite sequence of set and of complementation of sets. Thus if $A,B,C$ are Borel sets then $A\cup B\cup C$ is a Borel set, and if $A_1,A_2,A_3,\ldots$ is an infinite sequence of Borel sets then $A_1\cup A_2\cup A_3\cup\cdots$ is a Borel set.
By de Morgan's laws we have
\begin{align}
& A_1 \cap A_2\cap A_3\cap \cdots \\[8pt]
= {} & \mathbb R\smallsetminus\big((\mathbb R \smallsetminus A_1) \cup(\mathbb R\smallsetminus A_2) \cup(\mathbb R\smallsetminus A_3) \cup\cdots\big)
\end{align}
and so the intersection is also a Borel set.
Although the class of Borel sets is closed under the operations of union and intersection of infinite sequences of sets, it is not closed under arbitrary unions or arbitrary interesections. For example, the set of all open sets that contain $0$ as a member has an intersection (namely the singleton set $\{0\}$) but this operation of taking the intersection of a set of sets is not an instance of taking the intersection of a sequence of sets. (Nonetheless, the singleton set $\{0\}$ is a Borel set, since it is the complement of the open set $(-\infty,0)\cup(0,+\infty).$
Thus:

*

*Start with open sets.

*Close under the operations complementation of sets and of union of (finite or infinite) sequences of sets, and consequently (by de Morgan) also under the operation of intersection of sequences of sets.

What you get is Borel sets.
A probability distribution assigns a probability to every Borel subset of $\mathbb R.$
