Posted by Peyam
on November 9, 2007, 1:27 pm
I was just wondering about something:
What if we started measure theory by defining inner measures instead of outer measures.
More precisely, define the inner measure m*, which is a function from Rn to R such that
m*(O) = sup (Sum |Oi|), where the supremum is over all countable "fillings" of O (a "filling" is just the union (over i) of Oi, where each Oi C O.
Would we get anything different (ie theorems that cannot be proven with inner measure, but with outer measures, or theorems that cannot be proven with outer measures, but that can be proven with inner measures)
And what happens if the inner measures and the outer measures of O are equal? Does O have then a special property? (I'm being inspired by the fact that a function on R is Riemann integrable if the Lower Integral, using lower sums, and the Upper Integral, using upper sums, are equal. Perhaps we can do something in the general measure case as well!, ie O is Peyam measurable if the interior and exterior measure of O are equal)
Your Peyam
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