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>So if we have a bounded set (and I've not given a technical definition of what that means) then we'd like the measure of that to exist Why in the world should

by sjolsen 11y ago

>So if we have a bounded set (and I've not given a technical definition of what that means) then we'd like the measure of that to exist Why in the world should this be true? I'd much sooner try to weaken this requirement than the additivity requirement or the axiom of choice. For example, I imagine restricting measures to (unions of countably many disjoint) (path-)connected subspaces of the ambient topology would solve the problem just as handily, without making any compromises on how well the math models our intuitive notion of measure. On that note, I suspect that to really have a meaningful notion of measure, one really needs to have at least a topology, if not an outright metric space. Can anyone more versed in this area comment on this?

ColinWright 11y ago
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 ...
- tripzilch 11y ago
  > 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.
ebola1717 11y ago
A lot of the applications of measure theory are in fourier analysis and probability, where countable sums are necessary, and topology often isn't. Also, just because the topology on R yields a certain intuition doesn't mean it's the right way to generalize the theory.

by sjolsen 11y ago

ColinWright 11y ago

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 ...

tripzilch 11y ago
> 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.

ebola1717 11y ago

A lot of the applications of measure theory are in fourier analysis and probability, where countable sums are necessary, and topology often isn't. Also, just because the topology on R yields a certain intuition doesn't mean it's the right way to generalize the theory.