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: 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 $\mathbb R.$
A: I am not really an expert, not sure that will be able to explain coherently.
On a technical level, attributing a probability to any subset can be shown to be not realistic. This is somewhat technical - there are more subsets than subsets that can be reasonably described ("non-measurable subsets"). In some sense, one can show that there are subsets for which there is no description - this is something about axiomatics of mathematics. So one describes a huge collection of subsets of, say, $[0,1]$, which is closed under reasonable operations, and you would like to only ask about the probability of falling into such subsets (the collection is the $\sigma$-algebra).
But I think that there is also a less technical facet, more probability-related. It might be that you started with a system, and then consider experiments of some more restricted sort, and those experiments can never distinguish between two events $\omega_1$ and $\omega_2$. Then you would like, to adjust to this new reality, to have all your "legitimate" subsets of events to either contain both $\omega_1$ and $\omega_2$ or not contain neither. So you come up with the idea of a $\sigma$-algebra adjusted to this new reality, of a person with glasses of worse quality (so you can look up something like "tail $\sigma$-algebra" or something like this... not really an expert...)
A: It is better to correlate this with lebesgue measure.
Rieman integral fails coz it cannot break down the input after 3 dimensions. For 1d it’s rectangles...for 2d it’s cubes etc what about 4d?
By shifting domain from x to y we end up with dividing the different function outputs into slices and then multiplying them by input measure. This becomes a measure problem and assigning a measure has to obey some conditions.
Like for examples areas should be additive. Skipping over much detail here. If you use power set as sigma algebra then some of these basic conditions are not satisfied. Now if you read standard definition of sigma algebra it will make more sense. It’s just a way of assigning measure for use in lebesge integration
A: I will try to give my personal perspective on Borel–$\sigma$ algebras. Hopefully, it will be useful for both of us.
If we consider the Riemann integral, there are some natural questions we can ask. Suppose that $f_n$ are a sequence of continuous functions, for simplicity on $[0,1]$. We know that $f_n$ are integrable because they are continuous. Furthermore, let us suppose that $f_n \rightarrow f$ as $n\rightarrow \infty$. Now what can we say about the (Riemann) integrability of $f$ and what is $\int f$?
I will spoil the answers to these questions and say that we can not deduce anything about $f$, because we can find a sequence of bounded continuous functions which converge to a non–integrable (non Riemann integrable) function
That is besides the point. Let us pretend that we didn't know the answer above and try to investigate possible answers to the questions I raised earlier. Naturally, we need to understand what kind of function $f$ is before we attempt to address what $\int f$ is or even whether it is integrable.
Well, it turns out it is a Borel function. Meaning, $f^{-1}(A)$ is a Borel set for a Borel set $A  \subset \mathbb{R}$. To see why, notice that if $x \in f^{-1}((a,\infty)) $ then for some large $N_x$, we get for all $n\ge N_x$ that $x\in f^{-1}_{n}((a,\infty))$. So after some thinking, we would deduce that $$f^{-1}(a,\infty) = \bigcup_{n=1}^{\infty}\bigcap_{m=n}^{\infty}f_m^{-1}(a,\infty)$$
Since, $f_m^{-1}(a,\infty)$ is open set for all $m$, in order to understand $f$, we need to understand what happens when we take countable intersection of open sets. I am too tired to write about why complement is important but a similar heuristic can be provided. Maybe I will do that tomorrow.
A: Basically, a $\sigma$-algebra is a subset of all the possible sets X on space and allow's us to define a measure. Basically, how large is each subset of X provided it is in the $\sigma$-algebra. A Borel $\sigma$-algebra is a $\sigma$-algebra which contains all the open, closed, half-way open, half-way closed, and ray sets within a space. Consider, R. A Borel $\sigma$-algebra will have all the $(a,b), [a,b], (a,b], [a,b), (-\infty,a),(-\infty,a]$. At the high level, we can figure out how large each of the above sets is on a measure.
A: Say you have something random that produces a real number.
Say I can tell you the probability the number is greater than a, less than b, in between a and b inclusive etc. That seems pretty reasonable because saying the quantity is in between two numbers is useful for constraining it. Those are the nicer sets to work with.
Suppose also if I can tell you the probability for $X$ and $Y$, then I also can do $X \bigcup Y$ and $X \bigcap Y$ and $X \setminus Y$. Same if there are more than just 2 to take unions and intersections.
For example, if I know the probability for being between 2 and 3 is .1 and the probability for being above 6 is .001, then I can immediately say .101 for the union because it has to be exactly one of those two events because it is a disjoint union.
But even with all these sets I can tell you probabilities for, there are still lots of sets I cannot because they aren't built up from intervals. Intervals were my starting point of understanding this random number.
