It turns out that pre-measures give rise quite naturally to outer measures, which are defined for all subsets of the space " X ".
22.
If is a metric outer measure on, then every Borel subset of is-measurable . ( The Borel sets of are the elements of the smallest-algebra generated by the open sets .)
23.
The problem is that the function given by the lim sup formula is not countably subadditive in general and in particular is infinite on any set without compact closure, so is not an outer measure .)
24.
The first principle is based on the fact that the inner measure and outer measure are equal for measurable sets, the second is based on Lusin's theorem, and the third is based on Egorov's theorem.
25.
If for all possible such subsets A of the real numbers, the partitions of A cut apart by E have outer measures which add up to the outer measure of A, then the outer Lebesgue measure of E gives its Lebesgue measure.
26.
If for all possible such subsets A of the real numbers, the partitions of A cut apart by E have outer measures which add up to the outer measure of A, then the outer Lebesgue measure of E gives its Lebesgue measure.
27.
Their second formulation is as follows : for any measurable functions over that are linearly independent over any subset of of positive measure, there is a linear combination such that the surface, dividing into and, simultaneously bisects the outer measure of.
28.
Note that the image of such a set " N " is not necessarily complete, it follows that if the Lebesgue outer measure of that set is zero, then it is measurable and its Lebesgue measure is zero as well.
29.
An outer measure satisfying only the first of these two requirements is called a " Borel measure ", while an outer measure satisfying only the second requirement ( with the Borel set B replaced by a measurable set B ) is called a " regular measure ".
30.
An outer measure satisfying only the first of these two requirements is called a " Borel measure ", while an outer measure satisfying only the second requirement ( with the Borel set B replaced by a measurable set B ) is called a " regular measure ".
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