lebgues measure

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Property that proves most results in Lebesgue measure

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Proposition 13 : Lebesgue measure is countably additive

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Length of an Interval II lebesgue measure II Real Analysis

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Example of Outer measure

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proof that Measure of an empty set is zero

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For epsilon and outer Measure this inequality holds

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Example:- If M*(A)=0 then M*(AUB)=M*(B)

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Proposition 2 : Outer Measure is translation Invariant

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Proposition 6 : let A be any set and E be the finite disjoint collection measurable set then

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Example:- outer measure of irrational no in [0,1] is = 1

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Proposition 7 : Union of Countable collection of measurable set is measurable

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Measurable set II HINDI II MEASURE THEORY II

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Union of two Measurable set is measurable

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Proposition 12 :-E be any finite outer measure. then is disjoint collection of open intervals

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Monotonicity property proof

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E is measurable set iff this inequality holds II HINDI II

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Intersection of finite collection of measurable set is measurable

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Proposition 4 : Any set of outer measure is Zero is measurable II HINDII

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E is measurable then it's complement is also measurable

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Union of two measure set is measurable II HINDI

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Proposition 5 : Union of finite collection of measurable set is measurable II Hindi II

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Proposition 10 : - The translate of measurable set is measurable

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Example:- If E1 and E2 are measurable set then it's difference is also measurable

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Exicison property II HINDI II PUNE UNIVERSITY