Glivenko–Cantelli theorem
Imagine you have a big basket full of apples. You want to know if these apples are all ripe or not. You pick one apple, check if it's ripe or not and put it back into the basket. You repeat this process with all the apples in the basket. When you're done, you count how many ripe apples you found and divide that number by the total number of apples in the basket. This gives you the proportion of ripe apples in the basket.
Now, let's say you have another basket, but this one is much larger and has a lot more apples in it. You don't have time to check every single apple in the basket, but you still want to know what proportion of them are ripe. You decide to randomly pick a few apples from the basket, just like before, but this time, you're only going to check a small number of them. Let's say you check 10 apples and find that 8 of them are ripe.
The Glivenko-Cantelli theorem says that as you randomly pick more and more apples from the basket and count the proportion of ripe ones, that proportion will get closer and closer to the true proportion of ripe apples in the basket. In other words, even if you can't check every single apple in the basket, you can still get a pretty good idea of what proportion of them are ripe just by checking a few (but enough) random ones.
Now, let's say you have another basket, but this one is much larger and has a lot more apples in it. You don't have time to check every single apple in the basket, but you still want to know what proportion of them are ripe. You decide to randomly pick a few apples from the basket, just like before, but this time, you're only going to check a small number of them. Let's say you check 10 apples and find that 8 of them are ripe.
The Glivenko-Cantelli theorem says that as you randomly pick more and more apples from the basket and count the proportion of ripe ones, that proportion will get closer and closer to the true proportion of ripe apples in the basket. In other words, even if you can't check every single apple in the basket, you can still get a pretty good idea of what proportion of them are ripe just by checking a few (but enough) random ones.