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.
Best Answer
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.