Monotone convergence theorem (almost everywhere concept) Could someone enlighten me. Monotone convergence theorem says: if $f_{n},f\in L^{+}$ i.e measurable functions, such that $f_{n}\uparrow f$ pointwise then $\int f_{n}\uparrow \int f$.
But then my teacher says we can relax the condition, using almost everywhere.
We can assume that $f_{n}\uparrow f$ a.e $\Rightarrow \int f_{n}\uparrow \int f$ and then he does something I don't get. 
He asks the question:
''Is it true $f_{n}\to f$ a.e $\Rightarrow \int f_{n}\to \int f$?''
what does the horizontal arrow mean in this case? (does he also mean converge to but almost everywhere?)
Here is his counterexample 
$(X,\mathcal{M},\mu)=(\mathbb{R},\mathcal{L},m)$ and we let $f_{n}=\chi_{[n,n+1]}$ 
then $\int \chi_{[n,n+1]}dm=m([n,n+1])=1$ but $f_{n}\to 0$ pointwise
Is the horizontal and the vertical arrow meaning the same (''converge to'')?
Why do the characteristic functions converge pointwise??

The hypotheses can be weakened to being positive almost everywhere, increasing in $n$ almost everywhere, and converging pointwise almost everywhere, by replacing $f_n$, $f$ by 0 at points where either positivity, monoticity or pointwise convergence fails and using the results that the integrals of functions that agree a.e. are equal. I can explain more if you like.
In your example it's not increasing anywhere in $n$, which is why the MCT fails.
