The concept of a sigma algebra is particularly useful in analysis. It can actually be quite difficult to prove that any explicit set is (Lebesgue) measurable. However, the set of measurable sets forms a sigma algebra. Using this fact, we can get a whole lot of sets by simply proving that all open and closed sets are measurable. In particular, given those facts any Borel set is indeed measurable.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."