I want to find examples for
a) A sequence $(f_n)_{n\in \mathbb N}$ of non-negative, measurable functions that converge pointwise, but $$\int f_n \,d\mu$$ does not converge.
b) A sequence $(f_n)_{n\in \mathbb N}$ of measurable functions with $f_1 \leq f_2 \leq \dotsc$ with existing pointwise limit $f:= \lim_{n\to \infty} f_n$ but $$\int f_n \, d\mu \neq \int f\, d\mu$$
Attempt: a)
Define $f_n := \left ( 1 + \frac{1}{n}\right ) \mathbb 1_{[0,n]}$. Then $f_n \to 1$ pointwise but $$\int f_n(x) \, dx= n+1 \to \infty$$
Is this correct? Moreover, is it possible to find a function like this but we allow convergence in $\mathbb R \cup \{ \infty \}$, i.e. where the sequence of integrals must not have a limit in $\mathbb R \cup \{ \infty \}$?
b)
Define $f_n = -\frac{1}{n} \mathbb 1_{]-\infty, \frac{1}{n}]}$. Then $f_n \to 0$ pointwise, but $$\int f_n(x) \, dx = -\infty \neq 0 = \int 0 \, dx$$
again, is this correct and does there exists a solution with finite values? Thanks in advance!