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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?

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    $\begingroup$ A measure assigns numbers to subsets of $X$ (e.g. their volume), while a metric assigns numbers to pairs of points of $X$ (their distance). What resemblance do you see here that suggests a generalization? $\endgroup$
    – Christoph
    Aug 17 '20 at 10:19
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    $\begingroup$ Measurable sets are closed under complement; open sets are not. The two subjects have very different concerns. Beyond the level of "general abstract nonsense" it's not clear there's much to be said about the category of structures you have in mind that contains both measure and topological spaces. Members of this category have to be closed under countable unions & finite intersections, and not necessarily complement. Are there two nontrivial theorems, one of measure theory and the other of topology, that are just variants and have essentially the same proof? (Rhetorical question.) $\endgroup$
    – BrianO
    Aug 17 '20 at 10:27
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    $\begingroup$ The definitions of a topology and a $\sigma$-algebra may seem similar, but are different in practice. Most topologies you encounter aren't $\sigma$-algebras and vice versa. A collection of subsets that is both a topology and a $\sigma$-algebra is the entire power set as soon as it is $T_1$. So such a collection is either this one trivial example or rather ugly. $\endgroup$
    – Thorgott
    Aug 17 '20 at 10:47
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    $\begingroup$ First of all, make it clear whether you want to talk about analogies between $\sigma$-algebras and topologies or between measures and metrics, because those are different comparison. In non-discrete measure spaces, the measure of a pair of points will always be zero, so what you're suggesting doesn't really work. Now, you offer the alternative of defining the distance to be the measure of the line connecting them and this works, but only because we have a notion of straight line in vector spaces. This doesn't generalize to arbitrary measure spaces at all. $\endgroup$
    – Thorgott
    Aug 17 '20 at 11:04
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    $\begingroup$ On the other hand, in Riemannian geometry, there is the notion of geodesics (shortest paths) between two points on a Riemannian manifold and the length of a geodesic connecting two points (roughly speaking) serves as metric on the manifold that induces the topology one started with, so this could be a generalization of your idea. But still, measure spaces are much, much, much more general than Riemannian manifolds. $\endgroup$
    – Thorgott
    Aug 17 '20 at 11:06
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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?

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  • $\begingroup$ Your explanation provides lot of insight. The main reason I asked the question is to know as to why we could not just define a metric as a measure restricted to two points defined the way it is? That way, you have your definition of a metric but through the definition of a measure. While measure can enjoy more general forms, a metric could just be defined as a measure restricted to two points. Why do we another set of axioms for a topological space? After all, a metric is measuring distance between two points and so qualifies to be a measure right? $\endgroup$ Aug 17 '20 at 10:51
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    $\begingroup$ We might find a way to: 1. Define a line connecting two points, 2. Define an analog of the 1d Hausdorff measure on this space, and 3. Define the metric as the Hausdorff measure of the line connecting two points. But we would have to make sure that this is actually doable for every possible metric, and then the construction still comes with a lot of overhead. Also, not every topological space is metrizable, so this approach will not work for a general topological space. Sometimes it's just simpler to introduce more different structures than to force a given structure into a suitable behavior. $\endgroup$ Aug 17 '20 at 11:06
  • $\begingroup$ This is exactly what I guessed. It is perhaps possible to define a metric as a restriction of a measure on open balls, but then as you mentioned, this may not be always possible for various types of metrics and topological spaces. Sure enough, I need to gain more knowledge to appreciate these facts. $\endgroup$ Aug 17 '20 at 11:11
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    $\begingroup$ I think the sentence "Sometimes it's just simpler to introduce more different structures than to force a given structure into a suitable behavior." gets so succinctly to the heart of the 'right' answer to this question that it might be worth making it part of your answer. ('Right' in quotes because it's my subjective judgement, not because of scare quotes.) $\endgroup$
    – LSpice
    Aug 17 '20 at 21:14
  • $\begingroup$ @LSpice I suppose your short comment sums up the answer nicely :) $\endgroup$ Aug 18 '20 at 4:30

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