# Monotone Convergence for Decreasing Functions

I'm trying to prove the Monotone Convergence Theorem for decreasing sequences, namely if

Let $(X,\mathcal{M},\mu)$ be a measure space and suppose $\{f_n\}$ are non-negative measurable functions decreasing pointwise to $f$. Suppose also that $f_1 \in \mathscr{L}(\mu)$. Then $$\int_X f~d\mu = \lim_{n\to\infty}\int_X f_n~d\mu.$$

Why does this statement not follow from LDCT with $f_n$ being dominated by $f_1$?

I'm also aware of the solutions with $g_n=f_1-f_n$, but the question asks to prove it using Fatou's lemma

• In Fatou's lemma, the inequality goes the other way. So the inequality $\lim\int f_n\,d\mu\le\int f\,d\mu$ doesn't follow from Fatou. Oct 9, 2015 at 1:26
• @grand_chat why does the statement not follow from LDCT with $g=f_1$? Oct 9, 2015 at 1:59
• I think it does follow from LDCT.
– user99914
Oct 9, 2015 at 3:01
• @JohnMa Then why does every solution use the $g_n=f_1-f_n$ trick and MCT? Oct 9, 2015 at 3:03
• I guess that's because LDCT is an overkill? (It seems that LDCT is the last theorem to prove, either Fatou or MCT comes first)
– user99914
Oct 9, 2015 at 3:05