|WikiProject Mathematics||(Rated Start-class, Low-priority)|
How about countable additivity, isn't that forgotten?
hyphen or no hyphen?
Something is wrong with the definition. For example, let R be the family of all sets of reals which are either empty or contain 0, and let μ0 be an arbitrary function from R to [0,∞] such that μ0(∅) = 0. Then μ0 is a premeasure according to our definition: σ-additivity holds trivially because there are no disjoint nonempty sets in R. Clearly, it is not true in general that such an μ0 can be extended to an outer measure, for example this is impossible if the function μ0 is non-monotone. It seems to me that one should additionally require R to be closed under set difference to make it work.—Emil J. 16:25, 19 October 2012 (UTC)