I am struggling with an application of the Dominated Convergence Theorem (DCT) which has cropped up a few times in various proofs I have been studying, in particular a proof about approximating Lebesgue integrable functions by step functions that are Riemann integrable. The problem appears like it should be easy, but I struggle nonetheless! I would appreciate very much if somebody would make me feel silly and point out the steps I am missing.

Let the measure space be $(\mathbb{R},\mathscr{R},\lambda)$, where $\mathscr{R}$ is the sigma field of linear Borel sets, and $\lambda$ is Lebesgue measure. Suppose $f$ is Lebesgue integrable, and $f_{n}=fI_{[-n,n]}$. The conclusion of the proof is that is that $\int|f-f_{n}|dx\rightarrow 0$ using the DCT.

The proof is very brief, and says we have $f_{n}\rightarrow f$ (this to me is clear), and also that $f_{n}\leq |f|$ for all $n$, and so by the DCT we have the required result (this I cannot follow).

Firstly I am inclined to think that we have $|f_{n}|\leq |f|$, so that by the DCT we have $\int f_{n}dx\rightarrow\int fdx$. Is this correct? Even if this is so I still cannot get the final result. To use the Theorem directly I need to somehow show $|f-f_{n}|\leq g$ for $g$ integrable. Then since $|f-f_{n}|\rightarrow 0$, the result will indeed follow from the DCT.

Any help would be greatly appreciated.


We can use the triangle inequality to get the following bound on $|f-f_n|$.

$$ |f-f_n| \leq |f| + |f_n| \leq 2|f|, $$

where the last inequality follows because $|f_n|\leq|f|$. So we can take $g=2f$ and the conditions of the dominated convergence theorem are satisfied.


By request of the OP, I am explaining my use of the triangle inequality.

$$ |f-f_n| = \left|f+(-f_n)\right| \leq |f|+|-f_n| = |f| + |f_n| $$

This helps to expand the applicability of the triangle inequality, depending on the definition being used for the triangle inequality.

  • $\begingroup$ Using the triangle inequality to determine bounds for DCT applications is a very common tool, so it is good to practice these types of problems. $\endgroup$ – Carl Morris Oct 2 '12 at 18:14
  • $\begingroup$ Thank-you Carl for your very quick response. I know I am being very dense here but I did play around with the triangle inequality but as far as i can tell I can only use the backwards version: $|f-f_{n}|\geq||f|-|f_{n}||$. Can you humour me and show me how you get your version? Many thanks. $\endgroup$ – dandar Oct 2 '12 at 18:20
  • $\begingroup$ You're welcome, I hope the additional steps help. $\endgroup$ – Carl Morris Oct 2 '12 at 18:28
  • $\begingroup$ Of course, so simple! I did not realise we could use the triangle inequality like that. Thanks very much Carl. $\endgroup$ – dandar Oct 2 '12 at 20:24

Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.