Let f be a nonegative measurable function on R . Show that lim n → ∞ ∫ − n n f = ∫ R f Of course ∀ n , ∫ − n n f ≤ ∫ R fTo show the reverse, I feel that I need to use Fatou's lemma; but can't seem to get the right argument.

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Let   f be a nonegative measurable function on       R  . Show that                                          lim                          n              →              ∞                                            ∫                          −              n                                      n                                f          =                      ∫                                          R                                              f                    Of course  ∀  n  ,      ∫          −      n              n        f  ≤      ∫                  R              fTo show the reverse, I feel that I need to use Fatou&#039;s lemma; but can&#039;t seem to get the right argument.

tatovihd1 · Accepted Answer

The sequence of integrals is not your main concern. The sequence of functions       f    n    (  x  )  =  f  (  x  )  ⋅            1              (      −      n      ,      n      )       is your main concern.

tilfaen4a · Accepted Answer

You may apply the dominated convergence theorem with      ∫          −      n              n        f  =      ∫                  R              f  (  x  )  ⋅            1              (      −      n      ,      n      )        ≤      ∫                  R              fand you may conclude easily.

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