Talk:Outer measure

I am rewriting this entry in User:Camillo.delellis/sandbox. Camillo (talk) 18:12, 20 September 2012 (CEST)

Done. Camillo (talk) 20:43, 20 September 2012 (CEST)

"Outer measures induced by measures": It seems, $\mathcal{H}$ is used but not defined. True,in the next phrase something is said about $\mathcal{H}$, but only in context of "some authors"; is the matter clear? Also: induced by measures? or by just set functions? --Boris Tsirelson (talk) 21:47, 20 September 2012 (CEST)

In fact in a first version I had those two sentences swapped. Now it should be more clear. Indeed we could call the section Outer measures induced by set functions. However this construction seems more commonly used when you have a $\nu$ which is already a measure or at least a finitely additive set function: the idea of the title is to draw the attention on the second point of Theorem 6. Camillo (talk) 22:00, 20 September 2012 (CEST)

But on the other hand, first, who defines outer measure induced by a measure via covers? In this case, a much simpler definition is available. And second, Caratheodory construction does not start from a measure; and $\nu (A) = c_\alpha ({\rm diam}\, (A))^\alpha$ is not a measure. --Boris Tsirelson (talk) 07:45, 21 September 2012 (CEST)

OK. You conviced me. Camillo (talk) 08:42, 21 September 2012 (CEST)

"Remark 9": The Hausdorff measure, is it a measure or an outer measure? Later, in "Examples", it is claimed to be an outer measure; really? --Boris Tsirelson (talk) 21:53, 20 September 2012 (CEST)

Good point. I should call them all "outer measures" (even the ones in the examples) and specify that when the "outer" is dropped, then one refers to their restriction on the corresponding class of measurable sets. Camillo (talk) 22:00, 20 September 2012 (CEST)