In McKeans stochastic integrals from 1969 he proves this:
You have a filtered probability space $(\Omega,\mathcal{F},P)$, where the filtration is based on a Brownian motion. Assume that $X_t$ is a process that is jointly measurable ($\mathcal{B}(\mathbb{R})\otimes \mathcal{F}$), and adapted to $\mathcal{F}_t$, and also assume that $P(\int_o^T X_t^2 dt<\infty)=1$, then there exists a sequence of simple functions of the form $K_t^n=\Sigma_{1}^{N(n)}\phi_j(\omega)\mathcal{X}_{[t_j,t_{j+1})}$ ,where the simple functions satisfies adaptedness, joint-measurability and the same integrability requirement as $X_t$. And we also have that $P(\int_o^T(X_t-K_t^n)^2dt\,\,\,\,\,\, \overrightarrow{n \rightarrow \infty} \,\,\,\,\,\,0)=1$.
I am wondering if where I wrote $X_t^2$ and $(X_t-K_t^n)^2$, I could have written $|X_t|$ and $|X_t-K_t^n|$, does the statement of the theorem still hold?
I don't really understand the proof, so I am not able to generalize it, or see if it would work if p=1. Do you know?
I think it would follow from the above theorem if we had that if $f_n,f$ is positive, integrable and $\int_0^T|\sqrt{f_n(t)}-\sqrt{f(t)}|^2dt \rightarrow 0$, then we also have that $\int_0^T |f_n(t)-f(t)|dt\rightarrow 0$, but I can't see if this holds either. Do you see if this one holds?