Monotone Convergence Theorem for non-negative decreasing sequence of measurable functions

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 $\int f_1 \lt \infty$. Then $$\int_X f~d\mu = \lim_{n\to\infty}\int_X f_n~d\mu.$$

Atempt:

Since $\{f_n\}$ are decreasing, and converges pointwise to $f$, then $\{-f_n\}$ is increasing pointwise to $f$. So by the monotone convergence theorem $$\int_X -f~d\mu = \lim_{n\to\infty}\int_X -f_n ~d\mu$$ and so $$\int_X f~d\mu = \lim_{n\to\infty}\int_X f_n~d\mu.$$

• Your attempt is on the right track but is not quite right. In particular, you might think about the hypothesis $\int f_1 < \infty$ and whether you've used it. Hint: What do you know about $g_n = f_1 - f_n$? Mar 24, 2012 at 20:06
• @cardinal: oh yes....$g_n \geq 0$...Thanks
– Kuku
Mar 24, 2012 at 20:07
• Yes, $g_n \geq 0$...and, what else? Davide's answer lays out the details. (+1 for showing your work.) Mar 24, 2012 at 20:08
• Fair enough. Sorry, being a "standard" result, it sounded a bit like homework. Cheers. :) Mar 24, 2012 at 20:18
• Why not invoking the dominated convergence theorem with dominating function $f_1$? Aug 9, 2017 at 13:24

The problem is that $-f_n$ increases to $-f$ which is not non-negative, so we can't apply directly to $-f_n$ the monotone convergence theorem. But if we take $g_n:=f_1-f_n$, then $\{g_n\}$ is an increasing sequence of non-negative measurable functions, which converges pointwise to $f_1-f$. Monotone convergence theorem yields: $$\lim_{n\to +\infty}\int_X (f_1-f_n)d\mu=\int_X\lim_{n\to +\infty} (f_1-f_n)d\mu=\int_X f_1d\mu-\int_X fd\mu$$ so $\lim_{n\to +\infty}\int_X f_nd\mu=\int_X fd\mu$.
Note that the fact that there is an integrable function in the sequence is primordial, indeed, if you take $X$ the real line, $\mathcal M$ its Borel $\sigma$-algebra and $\mu$ the Lebesgue measure, and $f_n(x)=\begin{cases} 1&\mbox{ if }x\geq n\\\ 0&\mbox{ otherwise} \end{cases}$ the sequence $f_n$ decreases to $0$ but $\int_{\mathbb R}f_nd\lambda =+\infty$ for all $n$.
• @cardinal Nice proof, yet I have a question: You say that $g_n:= f-f_n$ is an increasing sequence of function, I understand this, but why is she positive? I've thought that since $f_n$ is decreasing to f than $f_n \ge f \forall n$ but this would mean that $f-f_n$ is negative. Where is my error? I can't find it Thank you in advance if you answer :-) Jan 17, 2014 at 7:53
• I used it with $f_1$ not $f$. Jan 17, 2014 at 10:17
• @Ale. $f_1\geq f_n$, since $f_n$ being decreasing and hence $f_1-f_n\geq 0$ for each $n$. Jun 30, 2016 at 5:44