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

Question

I&#039;m looking to show that       1                  |            x                        |                n             is not integrable on the subset of             R        n   where       |    x      |    ≥  1. It&#039;s easy to do on       R  , and I think I need to apply Fubini&#039;s theorem for the general case, but not sure how to do so in n-dimensional space. Thanks

knolsaadme · Accepted Answer

Via some computation and Tonelli&#039;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 &quot;surface measure&quot; on       S          n      −      1      . Now you can plug in   f  (  x  )  =      1                  |            x                        |                n                  χ          {      x      :              |            x              |            ≥      1      }       and compute it&#039;s integral.

I'm looking to show that 1 <mrow class="MJX-TeXAtom-ORD"> | <

Answered question

Answer & Explanation

New Questions in Pre-Algebra