I proved something that I don't think is true.
Let $f_n:X\to\mathbb{R}$ be a sequence of measurable functions that converges to $f$. Then if $\mu$ is a finite measure we have $$\lim \int f_n d\mu = \int f d \mu$$
For all $\epsilon>0$ there is a $N$ such that $n\geq N\implies$ $\vert f_n-f\vert<\frac{\epsilon}{\mu(X)}$. Then
$$\vert \int f_nd\mu - \int f\ d\mu\vert=\vert\int f_n-f \ d\mu\vert\leq\int\vert f_n-f\vert \ d\mu<\int \epsilon/\mu(X)\ d\mu=\epsilon$$ So it follows...
I don't think that is true because this was too easy and we just assumed that $\mu$ is finite... and at the other hand if $\mu$ isn't finite we must have the hypothesis of Lebesgue dominated convergence theorem (wich is much hard to prove)...
Is my proof good?