Let f n : R → [ 0 , ∞ ) be a sequence of measurable functions that converges to some function f. Suppose that I show that for "n big enough" we have | f n ( x ) | ≤ g ( x ) for all x ∈ R . Am I right in saying that I can still apply the DCT to conclude that lim n → ∞ ∫ R f n d λ = ∫ R f d λ ? I think that I can do this simply because I am removing a finite number of terms from my sequence ∫ R f n d λ and then I am applying DCT. Is this right?

Question

Let       f    n    :      R    →  [  0  ,  ∞  ) be a sequence of measurable functions that converges to some function f. Suppose that I show that for &quot;n big enough&quot; we have       |        f    n    (  x  )      |    ≤  g  (  x  ) for all   x  ∈      R  . Am I right in saying that I can still apply the DCT to conclude that             lim              n        →        ∞                    ∫                        R                            f      n        d    λ    =          ∫                        R                      f    d    λ  ? I think that I can do this simply because I am removing a finite number of terms from my sequence             ∫                        R                            f      n        d    λ   and then I am applying DCT. Is this right?

iskakanjulc · Accepted Answer

Suppose that       |        f    n    (  x  )      |    ≤  g  (  x  ) for all   n  ≥  N for some   N  ∈      N  . Let   (      h    n    ) be a sequence defined as       h    k    =      f          N      +      k      .Then in particular       h    k    (  x  )  ≤  g  (  x  ) for all   k  ∈      N  . Moreover you clearly have that       h    k    →  f when   k  →  ∞.So you can apply the DCT to (  (      h    k    )) and you obtain that      lim          k      →      ∞            ∫                  R                  h    k    =      ∫                  R              f  .Now if you go back to the definition of the limit : let   ϵ  &amp;gt;  0, the above limit implies that there exists   K  ∈      N   such that      |        ∫                  R                  h    k    −      ∫                  R              f      |    &amp;lt;  ϵfor all   k  ≥  K. This implies that      |        ∫                  R                  f          n        −      ∫                  R              f      |    &amp;lt;  ϵfor all   n  ≥  K  +  N by definition of       h    k  .So by definition of the limit, this gives you that      lim          n      →      ∞            ∫                  R                  f    n    =      ∫                  R              f  .

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