I am doing mathematical finance, and the authors always try to motivate the mathematical foundations of probability.
Often they roughly explain measure, and then talk about Lebesgue measure over, say, $[0,1]$. Then they make a statement like "the sigma algebra by beginning with the closed intervals and adding everything else we need to have a sigma algebra is called the Borel $\sigma$-algebra".
I'm never quite sure what that means. At the simples, since complements exist, we can get open intervals trivially using complement, intersection and union.
But I always feel like I am missing something in what I understand, or what is being explained. If we had the closed intervals with Lebesgue measure, why do we not have a sigma algebra? If things are "missing", exactly what are they, and is it always true we can add then with simply complement, intersection, and union? And, when we do so, is that the only sigma algebra that can come from closed intervals over $[0,1]$, or could others be built?