While reading this question here about the proof for Scheffe's Lemma, I was confused since someone said the proof in the question was not correct. I thought the argument was fine, and the author only needed to use the General Dominated Convergence theorem to finish the argument. Continuing form his/her work, we have that $$\lim_n \int f + f_n = \int 2f < \infty$$ which implies $\lim_n \int|f-f_n| = 0$.

General Lebesgue Dominated Convergence Theorem: Let $\{f_n\}$ be a sequence of measurable functions that converges pointwise a.e to $f$. Suppose there is a sequence $\{g_n\}$ of nonnegative measurable functions that converges pointwise a.e to $g$ and dominateds $f_n$ in the sense that $$|f_n| \leq g_n.$$

$$\text{If } \lim_n \int g_n = \int g < \infty, \text{ then } \lim_n \int f_n = \int f.$$

  • $\begingroup$ what is your question? do you mean the accepted answer to the question you linked? $\endgroup$ – Lost1 Jul 20 '14 at 15:36
  • $\begingroup$ @Lost1 Is my argument correct with the GLDCM? Because people in the other thread said that you can not use a sequence of functions as bounding function(s). $\endgroup$ – Xiao Jul 20 '14 at 15:37
  • $\begingroup$ I think your argument is fine but the way the OP in the other question wrote it is wrong. ( I have never seen this result you quoted before, but I just looked it up ) $\endgroup$ – Lost1 Jul 20 '14 at 15:41
  • $\begingroup$ I think the answer to that is not quite correct. it was given as a condition that $\int f_n\rightarrow\int f$ $\endgroup$ – Lost1 Jul 20 '14 at 15:43
  • $\begingroup$ @Lost1 OP in the other question didn't use the assumption $\int f_n \rightarrow \int f$ in order to apply the GLDCT, but his/her argument before was ok I believe. This is what I wanted to check. $\endgroup$ – Xiao Jul 20 '14 at 15:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.