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What is an unbounded set with finite measure?
by inimino 10y ago
What is an unbounded set with finite measure?
- obastani 10y agoI suppose you could take [0,1] \cup Z, and put a uniform measure on that set.
- vog 10y agoAs a somewhat extreme case, the rationals (Q) are unbounded but their measure is 0, hence finite.
- klodolph 10y agoSorry, by "finite" I also mean non-zero. You can't have a uniform distribution on a measure zero set.
- kutkloon7 10y agoYou can certainly have a uniform distribution on the set {1, 2}, which has measure zero... It seems to me that you can define a uniform distribution for finite sets of measure zero and for sets with finite measure.
- klodolph 10y agoOkay, again, I was being imprecise. All continuous uniform distributions have support with nonzero finite measure. The discrete uniform distribution is often thought of as a different distribution than the continuous uniform distribution. This just goes to show how much math relies on people knowing which definitions you happen to be using at the moment you say something. The point is that not all sets with nonzero finite measure are bounded, therefore you can have a uniform distribution whose support is equal to such a set.
- kutkloon7 10y agoI agree. I am of course nitpicking, but it is exactly the same mistake I made in the first post (but I was assuming a discrete distribution instead of a continuous distribution).
- klodolph 10y agoOther answers failed to give examples of a set with positive finite measure, which is what you really need. What we want is a set that has no upper bound, but isn't really weird or tricky—if it's too tricky, then the uniform continuous distribution doesn't make any sense—and an easy way to do this is take a union of a countable set of intervals. For example, consider a set that contains (0,1/2), (1,1+1/2), (2,2+1/4), ... etc, so interval i has measure 2^-i. It should be obvious that there is no largest element, and if you sum the measures of all the sets you will find that the measure is 1. A big chunk of "doing mathematics" is having a library of techniques on tap to come up with weird objects like this to attack assumptions.
- inimino 10y agoThanks. That seems obvious now that you point it out. Just like Zeno's paradoxes, which are also about stretching a finite measure over a countable infinity.