The packing measure of rectifiable subsets of the plane

It is usual to define Lebesgue outer measure in ℝ by using economical coverings by a sequence of open intervals. We start by outlining a different definition which gives the same answer for a bounded measurable E ⊂ ℝ. Put Then λ0 defines a pre-measure, but is not an outer measure because it is not countably sub-additive. However it leads to an outer measure by defining and it can be proved directly that λ is just another definition of Lebesgue outer measure. We do not give the details of this proof as it can also be deduced as a corollary of the main results in the present paper.