Let f n ∈ L 1 be a seq of nonnegative functions s.t f n → f a.e ptwise. Show lim n → ∞ ∫ ( m i n ( f , f n ) d μ = ∫ f d μ.For this, I wanted to use the dominated convergence theorem.I have shown that m i n ( f , f n ) converges to f pointwise since | f − f n | ≥ | f − m i n ( f , f n ) | .Moreover, | m i n ( f , f n ) | = m i n ( f , f n ) ≤ f. But I cannot use the dominated convergence theorem directly since the integral of f may not be finite. Is there a way to fix this?

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Let       f    n    ∈      L          1       be a seq of nonnegative functions s.t       f    n    →  f a.e ptwise. Show       lim          n      →      ∞        ∫  (  m  i  n  (  f  ,      f    n    )  d  μ  =  ∫  f  d  μ.For this, I wanted to use the dominated convergence theorem.I have shown that   m  i  n  (  f  ,      f    n    ) converges to   f pointwise since       |    f  −      f    n        |    ≥      |    f  −  m  i  n  (  f  ,      f    n    )      |  .Moreover,       |    m  i  n  (  f  ,      f    n    )      |    =  m  i  n  (  f  ,      f    n    )  ≤  f. But I cannot use the dominated convergence theorem directly since the integral of   f may not be finite. Is there a way to fix this?

g2joey15 · Accepted Answer

You will need more assumptions for DCT to work. For example suppose   μ is the Lebesgue measure on       R  . Now let       f    n    =      1          [      −      n      ,      n      ]       be the indicator function for the set   [  −  n  ,  n  ].       f    n   is       L    1   and       f    n    →  f  ≡  1 point wise. Hence DCT will not always work.Instead you want to use the monotone convergence theorem. Specifically, define      g    n    (  x  )  =      inf          k      ≥      n        min  (  f  (  x  )  ,      f    k    (  x  )  )  .Now       g    n   is an increasing positive sequence and       g    n    →  f. So we have that  ∫      g    n    d  μ  →  ∫  f  d  μ  .Finally we see that for each   n  ∫      g    n    d  μ  ≤  ∫  min  (  f  ,      f    n    )  d  μ  ≤  ∫  f  d  μThus, we have   ∫  min  (  f  ,      f    n    )  d  μ  →  ∫  f  d  μ.

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