Let ( X , M , μ ) be a measueable space, and let f : R → R be nonnegative f ( x ) ≥ 0 and measurable. Let E n = { x ∈ R : f ( x ) ≥ 1 n }. Show the following result ∫ R f d m = lim n → ∞ ∫ [ − n , n ] f d m = lim n → ∞ ∫ E n f d m .I was thinking maybe we can assume f χ E n ≤ f, monotone increasing sequence of nonnegative measurable functions, and then apply monotone convergence theorem, but I am not sure that we can find such increasing sequence, as 1 n is decreasing.

Question

Let   (  X  ,      M    ,  μ  ) be a measueable space, and let   f  :      R    →      R   be nonnegative   f  (  x  )  ≥  0 and measurable. Let       E    n    =  {  x  ∈      R    :  f  (  x  )  ≥      1    n    }. Show the following result      ∫                  R              f  d  m  =      lim          n      →      ∞            ∫          [      −      n      ,      n      ]        f  d  m  =      lim          n      →      ∞            ∫                  E        n              f  d  m  .I was thinking maybe we can assume   f      χ                  E        n              ≤  f, monotone increasing sequence of nonnegative measurable functions, and then apply monotone convergence theorem, but I am not sure that we can find such increasing sequence, as       1    n   is decreasing.

Zayden Andrade · Accepted Answer

Both sequences  f      χ                  E        n              ,  f      χ          [      −      n      ,      n      ]      are increasing because   f is non-negative,       E    n    ⊂      E          n      +      1      , and   [  −  n  ,  n  ]  ⊂  [  −  (  n  +  1  )  ,  n  +  1  ]. Now using monotone convergence theorem we have      lim          n      →      ∞            ∫                  E        n              f  d  m  =      lim          n      →      ∞            ∫                  R              f      χ                  E        n              d  m  =      ∫                  R                  lim          n      →      ∞            (    f          χ                        E          n                      )    d  m  =      ∫          R        f  d  mand similarly      lim          n      →      ∞            ∫          [      −      n      ,      n      ]        f  d  m  =      lim          n      →      ∞            ∫                  R              f      χ          [      −      n      ,      n      ]        d  m  =      ∫                  R                  lim          n      →      ∞            (    f          χ              [        −        n        ,        n        ]              )    d  m  =      ∫          R        f  d  m     .

Misael Matthews · Accepted Answer

Thank you, that&#039;s helpful!

Let ( X , <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="s

Answered question

Answer & Explanation

New Questions in Pre-Algebra