Definition of Outer measure II HINDI II MEASURE THEORY

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   • Example of Outer measure  

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Let I be a nonempty interval of real numbers. We define its length, £(1), to be 00 if I is
unbounded and otherwise define its length to be the difference of its endpoints. For a set
A of real numbers, consider the countable collections (h}~l of nonempty open, bounded
intervals that cover A, that is, collections for which A ~ U~l h. For each such collection,
consider the sum of the lengths of the intervals in the collection. Since the lengths are positive
numbers, each sum is uniquely defined independently of the order of the terms. We define
the outer measure3 of A, m*( A), to be the infimum of all such sums, that is
It


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Let I be a nonempty interval of real numbers. We define its length, 1(I), to be oo if I is
unbounded and otherwise define its length to be the difference of its endpoints. For a set
A of real numbers, consider the countable collections {Ik'1 of nonempty open, bounded
intervals that cover A, that is, collections for which 1 Ik. For each such collection,
consider the sum of the lengths of the intervals in the collection. Since the lengths are positive
numbers, each sum is uniquely defined independently of the order of the terms. We define
the outer measure3 of A, m* (A), to be the infimum of all such sums, that is
m*(A) = inf I(Ik)
1k=1

It follows immediately from the definition of outer measure that m* (0) = 0. Moreover, since
any cover of a set B is also a cover of any subset of B, outer measure is monotone in the
sense that
if ACB, then m*(A) m*(B).