Propostion 3 : Outer measure is countably subaddtive II HINDI II MEASURE THEORY II

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Inequality as mentioned in video :-    • For epsilon and outer Measure this in...  

As outer measure is countably subaddtive
If E k is the countable collection of set
From each Ek we find outer measure
Take union of all Ek and then it's measure is less than or equal to it's sum of each outer measure of Ek
To prove this we consider two cases
i) Suppose IF one of the Ek has infinite outer measure then the inequality holds trivally
ii) suppose each Ek has finite outer measure then by definition of outer measure there is the sequent of I k,i of open bounded interval which covers E k
For each Ek there is a Ik,i sequence which covers each Ek
If we take run both I and k then it's covers Uionn of Ek
The Lebesgue outer measure has a very nice property known as countable subadditivity. If we have a countable collection of subsets of , say , then the Lebesgue outer measure of the union will be less than or equal to the sum . ... Theorem: Let be a sequence of subsets of .

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