I am not going to assume a lot of technical backround for anyone reading this. For me, Mathematics is a subject about finiteness and how to define them followed by infinities and how to avoid them. So we will strip all unnecessary collections of abstract objects and build our intuition from the atoms.

Sets: atoms and thier collections

A set is just a collection. Finite sets are those for which we can count the number of elements in them. Infinite sets are those for which we cannot count the number of elements in them. Since we are physicists, we will not worry about the technicalities of infinite sets and will only focus on finite sets.

But why build a set at all in the first place. Sometimes, we do not collect the elements, but define a specific property. Anything that satisfies the property is an element of our set. This helps us because then it becomes easy to state what we mean by a certain set. For example, the set of all even numbers is defined by the property that a number is even if it is divisible by 2. So the set of people going to office everyday from 9 to 5, the set of all physicists that are poor in proving theorems, etc.

$$S= {x| \text{the set of people reading this post}} \text{is a small and finite set}.$$

From a set of points to a set of sets

What does measure theory want to measure?

A good way to define collecion of sets before we can measure things using them

Measurable functions

On convergence of these entities

And their world is pretty complicated