Motivation behind Borel $\sigma$ Algebras I understand what a Borel sigma algebra is, I am just not sure why we have the motivation to find such a sigma algebra.
It seems clear to me that the power set would satisfy everything I have learned about thus far.
At the moment I try to gather some intuition from topology. If we look at the box topology and the product topology on an infinite set. It turns out that the box topology simply has too many open sets in it for it to describe anything in a useful way.
I assume that somehow the power set simply has too many open sets in it for it be useful. To borrow a term from topology the Borel sigma algebra is defined in such a way that it is the coarsest sigma algebra that contains the things we are interested in studying my question is WHY do we need it.
 A: The born of the concept of $\sigma $-algebra is essentially due to two facts:

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*Our intuitive notion of what properties measurable objects must have

*The discovering that, in general, measures cannot be defined in the power set of $\mathbb{R}$
About the first point: we expect of measurable objects that if we have two unrelated objects that can be measured, then the total measure of both will be the sum of it measures, e.g. the mass of two marbles must be the same than the sum of it masses measured independently. Then we expect to extend this property to infinitely many objects, and we restrict ourselves here to infinite countable many objects (this agree with our notion of different objects in the space where each object have some dimension, they are not just "points", and so at most they can be infinite countable). From here we can develop the theoretical notion that we need for measurable objects giving a $\sigma $-algebra, if we understand that an object is represented by a subset of some set.
About the second point: the more important measure, the Lebesgue measure that have our intuitive notions of length, area or volume, show us that we cannot define it in the power set of $\mathbb{R}$, this is just not possible in the standard mathematical model we use, ZFC. This is why we care about $\sigma $-algebras, we cannot just assume the power set. There is also something more important: in practice a specific $\sigma $-algebra (generally a lot shorter from the ideal) born from the fact that every measure is restricted by the sensibility and precision of each sensor.
For the specific case of a Borel $\sigma $-algebra, it is because we want that the preimages of intervals of continuous functions will be measurable subsets, as we generally try to model everything continuously (when possible) and our measures are generally intervals or points.
You will start discovering that, in general, all abstract mathematical notions are almost natural and born from our most basic intuitions about our living experience. It is in the developing of the consequences of our abstractions where it lose the connection with it mundane history.
