People I talk to about this are often very unhappy about countable additivity, and are often unhappy about infinite in general. I suspect that the reason they think they want a measure to be defined on every set is because they don't know just how weird sets can be, and are relying on their intuition. Everything can be weighed, everything can be dunked in water to test its volume, so why should something not have a "volume"?
You clearly have a different intuition born of your background and training. You're talking about
> "... unions of countably many disjoint
> (path-)connected subspaces of the ambient
> topology ..."
You already have an unusual intuition.
And you're right - it's "obviously" better to restrict the sets we play with rather than limiting ourselves to finite collections, but that "obvious" comes from years, perhaps decades, of playing with these ideas.
And with regards having a topology or a metric, consider that perhaps a measure can be coerced into providing a metric ...
> Everything can be weighed, everything can be dunked in water to test its volume, so why should something not have a "volume"?
But it seems pretty obvious to me that you can't measure the volume of something (accurately) if it has features smaller than a water molecule.
Don't know about an equivalent example for weighing things.
But it stands to reason that, whatever you (think you) want to measure shouldn't have meaningful features smaller than the accuracy of the thing you're measuring with.
That's why we made electron microscopes, because photons were too big.
You clearly have a different intuition born of your background and training. You're talking about
You already have an unusual intuition.And you're right - it's "obviously" better to restrict the sets we play with rather than limiting ourselves to finite collections, but that "obvious" comes from years, perhaps decades, of playing with these ideas.
And with regards having a topology or a metric, consider that perhaps a measure can be coerced into providing a metric ...