Distributions are too wiggly to be functions. Is there a…
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Distributions are too wiggly to be functions. Is there a similar set of generalized functions that "aren't wiggly enough"?
Distributions let you rigorously discuss things like the delta function, or the derivative of the weirstrass function, even if they're too "wiggly" to be functions. The "too wiggly" part can essentially be summed up in them having nonzero "integrals" over arbitrarily small sets.
I wonder if there's a similar concept in the other direction. Rather than being so wiggly that they have nonzero integrals over arbitrarily small regions, can we have functions that are so "smooth" that they integrate to 0 over compact regions, but to nonzero values over infinite regions?
The default example I guess I'm going for is a "uniform probability distribution over the reals". Ideally, within whatever space we've defined, this would be the limit of wider and wider gaussians, just like how a delta distribution is the limit of taller and taller gaussians.
Maybe something like this could be achieved as continuous linear functionals on some other space of test functions? Another option would maybe be measures where you don't require countable additivity, just finite additivity?
I would love to hear everyone's thoughts.
#science
Distributions let you rigorously discuss things like the delta function, or the derivative of the weirstrass function, even if they're too "wiggly" to be functions. The "too wiggly" part can essentially be summed up in them having nonzero "integrals" over arbitrarily small sets.
I wonder if there's a similar concept in the other direction. Rather than being so wiggly that they have nonzero integrals over arbitrarily small regions, can we have functions that are so "smooth" that they integrate to 0 over compact regions, but to nonzero values over infinite regions?
The default example I guess I'm going for is a "uniform probability distribution over the reals". Ideally, within whatever space we've defined, this would be the limit of wider and wider gaussians, just like how a delta distribution is the limit of taller and taller gaussians.
Maybe something like this could be achieved as continuous linear functionals on some other space of test functions? Another option would maybe be measures where you don't require countable additivity, just finite additivity?
I would love to hear everyone's thoughts.
#science
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