Possible Duplicate:
Lp space convergence
This is related to a previous question that I posted. Here is the problem:
Suppose that $\{f_n\}$ is a sequence of functions on $[0,1]$ such that $f_n(x) \rightarrow f(x)$ almost everywhere. Suppose also that $\sup{\lVert f_n\rVert_{L^4}} = M < \infty $.
Prove that $\lVert f_n - f\rVert_{L^3} \rightarrow 0$.
I could complete this proof if I knew that $\lVert f_n\rVert_{L^3} \rightarrow \lVert f\rVert_{L^3}$ by a problem that I proved from Royden's book. How can we recover this fact from the problem statement?