Antiderivative of lebesgue integrable function Let f : [ 0 , b ] &#x2192;<

ntaraxq 2022-07-04 Answered
Antiderivative of lebesgue integrable function
Let f : [ 0 , b ] R be Lebesgue integrable. We define
g ( x ) = x b f ( t ) t d t , 0 < x b .
Show that g(x) is Lebesgue integrable in [0,b]. Also show that
0 b g ( x ) d x = 0 b f ( t ) d t .
If we show the asked equality then it is obvious that g is Lesbegue integrable, since f is. Now, I have try to show the equality using the fact that,
0 b [ g ( x ) f ( x ) x ] d x = 0, since g is the antiderivative of f but I had no success. I am pleased to know other ideas to approach this problem.
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Answers (1)

thatuglygirlyu
Answered 2022-07-05 Author has 14 answers
Explanation:
Let Δ = { ( x , t ) 0 x t b }. Define h(x,t) on [ 0 , b ] × [ 0 , b ] by h ( x , t ) = χ Δ ( x , t ) f ( t ) t . Apply Fubini's theorem to h to integrate it in two different ways. (There will be an initial step involving |h| to prove integrability of h.)
Edit: Δ would more accurately be changed in order to be a subset of ( 0 , b ] × ( 0 , b ]

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