Hyperfinite equivalence relation

Okay, imagine I have a big bowl of candy. Now, I want to sort them into different groups. I'm going to make a rule: any candies that taste the same will be in the same group. So all the chocolate candies will be in one group, all the fruity candies will be in another group, and so on.

This is kind of like how we make groups of things that are similar in math, too. We call these groups "equivalence classes."

But what if I had an infinite amount of candy? That would be a lot harder to sort!

That's where hyperfinite equivalence relations come in. It's kind of like a fancy trick we use in math to sort really big groups of things.

Instead of trying to sort all of the candies at once, we break them down into smaller groups. We keep breaking them down until we have a bunch of smaller, more manageable groups that we can sort easily based on their common features.

This idea of breaking things down into smaller groups is called "partitioning," and the smaller groups we end up with are called "finite equivalence classes."

So, a hyperfinite equivalence relation is just a really fancy way of saying that we can sort an infinitely large group of things into smaller groups of things that are similar, by using the idea of partitioning.