Why has there been no unification of topology axioms and measure theory axioms? Related thread here.
The axioms of a topological space and a measure space at the outset seem very similar. They differ in the closure axioms of unions and intersections. The uncanny resemblance between a metric and a measure makes me wonder as to why these axioms have been defined separately. Couldn't they develop a theory with just the concept of a measure and a measure space?
The one issue I see is that it might create circular logic. If we need topological space axioms to develop concepts in measure theory, that is a reason why we'd need to separate the two concepts. Closure of arbitrary unions versus countable unions, and finite intersections versus countable intersections, is not something I'd like to see as the only difference between the two concepts. Why have two separate systems when they are, at least from the outset, very similar concepts?
 A: Topologies and $\sigma$-algebras are designed with different objectives in mind. $\sigma$-algebras are designed to play nicely with measures, which are a generalized kind of volume measuring map. Topologies are designed to capture a notion of "closeness": when is a point $x$ close to a set $S$? If every open neighborhood of $x$ intersects $S$. When does a sequence get arbitrarily close to $x$? If every open neighborhood of $x$ contains points in the sequence. Stuff like that. So it's not surprising that at the outset, topologies and $\sigma$-algebras are different.
But! If we think about it some more, then we might find that intuitively, the open neighborhoods of a point are those which have a certain volume. Like, if I put an open ball around $x$, I can tell that it has a non-zero volume. And $\sigma$-algebras are designed to allow volume measurements. So shouldn't all the open sets somehow be made into a $\sigma$-algebra? After all, it might come in handy to assign a volume to such sets. And the answer is yes, that makes sense. We would like it a lot if we could assign a volume to open sets. For instance, this would allow continuous functions to play nicely with volume, since continuous functions play nicely with open sets. And that's why we define the Borel $\sigma$-algebra: given a topological space $(X,\tau)$, we define the Borel $\sigma$-algebra on $X$ as $\mathcal B(X):=\sigma(\tau)$, that is the smallest $\sigma$-algebra containing all the open subsets of $X$, so all the subsets which should have volume. Now $(X,\mathcal B(X))$ is a measurable space on which we could define a measure $\mu$ to assign a volume to each open set, if we were so inclined. This approach is often taken to define the Lebesgue measure, for instance. We take each open set of $\mathbb R^n$ and assign it the volume it should intuitively have, and then we take all the other sets we might get by uniting and intersecting these and assign them a volume which is in line with the definition of a measure. (There is a "better" approach using outer measures which yields more measurable sets, but this one is simpler.)
But the Borel $\sigma$-algebra is just one specific $\sigma$-algebra we might want. For other applications, different ones might work better, especially if we don't actually care about a sense of closeness on the underlying set. Then we don't need a topology, so why restrict our $\sigma$-algebra with a topology?
