Hi guys. I have one question regarding convergence in Lp. Suppose gn converges to g mu-almost everywhere on [0,1]. Suppose further that [tex]\left\|[/tex]gn[tex]\left\|[/tex]p[tex]\rightarrow[/tex]M < [tex]\infty[/tex]. How do I show that the pointwise limit g is in Lp?
So far, I know this is not true if we only know that gn goes to g mu-almost everywhere. I just don't see how the additional condition that the Lp-norms converge to some finite number implies that the limiting function g is also in Lp.