Weak convergence in $L^p$ equivalent to pointwise almost everywhere convergence Can weak convergence of a sequence $f_n\in L^p(\Omega, \mu)$ to some $f\in L^p(\Omega, \mu)$ be characterised as almost everywhere pointwise convergence?
Let us also assume the measure space is $\sigma$-finite.
I'm asking because for continuous functions on the unit interval, weak convergence is equivalent to pointwise convergence.

It isn't true.
The standard counterexample is to look at $L^2((0,1))$ with Lebesgue measure and take $f_n(x) = \sqrt{2} \sin(n \pi x)$.  The functions $f_n$ are orthonormal in $L^2$, so by Bessel's inequality they converge weakly to 0.  But pointwise, the sequence $\{f_n(x)\}$ diverges for every $x \in (0,1)$.
