About assumptions in the monotone convergence theorem

Question

Why is the hypothesis that $\left\{f_n \right\}$ be an increasing sequence essential to the monotone convergence theorem? Could someone provide a nice, easy to understand counterexample if I were to assume otherwise? Thank you.

Answer 1

Take for instance $f_n = \frac{1}{n}$, $f_n$ are deacreasing.

Then $\lim _{ n \rightarrow \infty} \int_{-\infty}^{\infty } f_n(x) \rm{d}x = \infty$

But $ \int_{-\infty}^{\infty }\lim _{ n \rightarrow \infty} f_n(x) \rm{d}x =0$

Answer 2

Here is a counterexample where $(f_{n})_{n\in\mathbb{N}}$ still converges a.e. to $f$: take $f=0$ and $f_{n}=n\mathbf{1}_{[0,n^{-1}]}$ on $[0,1]$. Then $f_{n}\to f$ everywhere but $0$, but $\int_{[0,1]}f(x)\,\mathrm{d}x=0$ while $\int_{[0,1]}f_{n}(x)\,\mathrm{d}x=1$ for all $n\in\mathbb{N}$. Requiring $(f_{n})_{n\in\mathbb{N}}$ to increase to $f$ prevents this kind of behavior (since all functions are assumed nonnegative).