A basic question on $L^p$ norm

Question

Consider a probability space and $f_m$ be sequence of measurable functions a.s. converging to $f$. What can be said about the limit

$$ \lim_{m\to \infty} \|f_m\|_m$$

where $\|.\|_p$ stands for the $L^p$ norm?

I don't want answer. Hint Enough.

Answer 1

Anything can happen. Take $f_m:=c_m\chi_{(0,a_m)} $ where $a_m\to 0$ and the unit interval is endowed with the Lebesgue measure.

• If $c_m=C a_m^{-1/m} $, the limit is $C$;
• if $c_m=m a_m^{-1/m}$, the limit is infinite;
• if $c_m=(-1)^ma_m^{-1/m}$, the limit does not exist.