Let $f \in L^1( [0,1])$. Show that
a) the limit of $||f||_p$ exists as $p \to 0^+$
b) If the measure of the set of points where $f$ is zero is positive, i.e. $ m( \{x\in [0,1] \mid f(x) = 0 \} ) > 0$, then the above limit is zero
I am trying to figure out how a) makes any sense since the function $f$ is in $L^1$. For $p \to 0 ^+$, does it mean that $ ( \int |f|^p) ^{1/p} < \infty$ ?