MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Let a ∈...

Wade Bullock

Wade Bullock

Answered

2022-07-10

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Let a X and μ be a measure defined on 2 X by
μ ( E ) = { 1 , a E 0 , a E
Write a simple necessary and sufficient condition on the non-negative functions f that ensures that
X f d μ <
My attempt:
We know that X f d μ = sup { X s d μ = y i μ ( A i ) s  simple  , s 0 , s f } , so we want the supremum to be finite and for this I'm thinking that the functions should have a finite border but I'm not sure if it works!

Answer & Explanation

Tanner Hamilton

Tanner Hamilton

Expert

2022-07-11Added 12 answers

y i μ ( A i ) = y i if there i is such that a A i and 0 if there is no such i. Note that if A i 's are disjoint there can be at most one i for which A A i Hnece s d μ = s ( a ). From this it follows that f d μ = f ( a ) for all non-negative measurable functiosn f. Hence, the conditioin is f ( a ) < .

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