Convergence almost everywhere and convergence in measure Let $(\mathbb{R},\mathcal{L},m)$, let $f_{n}(x)=n\chi_{[0,\frac{1}{n}]}$ then the sequence converges to $0$ everywhere except at $x=0$ thus $f_{n}$ converges a.e.
Then in my book (Folland) we have that if $f_{n}\to f$ a.e and $|f_{n}|\le g\in L^{1}$ then $f_{n}\to f$ in $L^{1}$ too. The above conditions are met.
Finally by another proposition we have that if $f_{n}\to f$ in $L^{1}$ then $f_{n}\to f$ in measure. 
I wonder if these relations holds for both finite and infinite measurable spaces? 

This works always, for example $(\mathbb R, \mathcal B, \lambda)$ forms a $\sigma$-finite but infinite measure space on which this works.
