I'm looking to show that 1 | x | n is not integrable on the subset of R n where | x | 1. It's easy to do on R , and I think I need to apply Fubini's theorem for the general case, but not sure how to do so in n-dimensional space. Thanks

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Answer & Explanation




2022-06-28Added 16 answers

Via some computation and Tonelli's theorem, we have for measurable f : R n [ 0 , ] that
R n f ( x ) d x = S n 1 0 f ( r ω ) r n 1 d r d S ( ω ) ,
where d S is "surface measure" on S n 1 . Now you can plug in f ( x ) = 1 | x | n χ { x : | x | 1 } and compute it's integral.

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