Example of Outer measure

What is cover:-    • Cover :- How union of open and bounde...  

What is length :-    • Length of an  Interval II lebesgue me...  

Definition of outer measure:-   • Definition of Outer measure II HINDI ...  


What are the example of outer measure
Example on outer measure
Hindi explanation of example on
outer measure


#outermeasure
#measuretheory
#exampleofoutermeasure
#lebsguemeasure
#realanalysis
#skclasses
#puneuniversity
#examples_of_outermeasure
#kthmcollege
#kthm
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 A C U' 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
ACCUHIk .
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
ifACB, then m*(A) m*(B).