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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.

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1 Answer 1

up vote 3 down vote accepted

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.

EDIT:

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.

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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. –  Carl Morris Oct 2 '12 at 18:14
    
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. –  dandar Oct 2 '12 at 18:20
    
You're welcome, I hope the additional steps help. –  Carl Morris Oct 2 '12 at 18:28
    
Of course, so simple! I did not realise we could use the triangle inequality like that. Thanks very much Carl. –  dandar Oct 2 '12 at 20:24

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