pointwise convergence and $L^1$ norm-boundedness implies what...?

Question

I have the following questions:

Suppose that $\mu$ is a probability measure on $R^d$, and that $f_n \to f$ a.s., and $\sup_n \int|f_n|d\mu \le C$ for some $C>0$. What can we deduce? For example: (it is ok to take some subsequence if necessary)

1. $\int |f_n - f |d\mu \searrow 0$? If not, then
2. $\int f_n d\mu \to \int f d\mu$? Still not, then
3. $\liminf_n \int f_n d\mu \ge \int f d\mu$? Still not, then
4. $\limsup_n \int f_n d\mu \ge \int f d\mu$?

or is nothing above true?

Answer 1

1. and 2. do not hold when $d=1$ and $\mu$ is the Lebesgue measure restricted to the unit interval. Indeed, let $f_n:=n \mathbf 1_{(0,1/n)}$. Then $ \int \left|f_n\right| \mathrm d\mu=\int f_n \mathrm d\mu=1$ and $f_n \to 0$ almost everywhere. 3. holds by Fatou's lemma if the functions are non-negative. If not,, then take $f_n:= -n \mathbf 1_{(0,1/n)}$ to show that 4. does not hold.