Essay
Introduction to Measure Theory
Frederic Schuller provides this wonderful explanation 1 of why the Borel sigma algebra is so interesting.
So only a topological space can carry a Borel sigma algebra because the whole idea of the Borel sigma algebra is that you employ the open sets in order to generate it. Okay, so you can do this, the question is whether this is a particularly good idea; it turns out to be a brilliant idea, okay? So again, why would one do this in the first place? Of course, you could take a topological space and establish a sigma algebra on it that has nothing to do whatsoever with the topology. That's perfectly fine.
However, you could also have a Hilbert space with an inner product, and you could establish ... a norm that has nothing to do with the inner product. You could do that, but ... you will not get the Cauchy-Schwarz inequality where on one side you have the inner product, on the other side you have the norms, because they have nothing to do with each other.
So usually, if you have a strong structure that is able to imply another structure, you will establish that other structure in such a way that it's being induced.
Now let's start with Hilbert space. You have an inner product, from that you induce the norm. From the norm, you would induce the standard topology by defining the soft balls using the norm. Once you have the topology, you induce the Borel sigma algebra ... so you only made one choice, namely an inner product, which is a very strong choice, right?
If you made a strong choice, usually you derive the weaker structures from that. Again, why would you do that? You don't have to. Well, because then you get the stronger theorems for the relation between these structures, right? So this follows standard procedure in mathematics to induce from given structures you have already chosen to not make yet another choice, but to induce it.