Be $1\leq p \leq \infty$. If $(f_{n})_n$ is a cauchy sequence in $\mathcal{L}^p(E)$ that converge pointwise to $f$ almost everywhere , show that $f\in \mathcal{L}^p$ and $\displaystyle \lim_{n \to \infty}{ N_{p}(f_{n}-f)}=0$
thx
Be $1\leq p \leq \infty$. If $(f_{n})_n$ is a cauchy sequence in $\mathcal{L}^p(E)$ that converge pointwise to $f$ almost everywhere , show that $f\in \mathcal{L}^p$ and $\displaystyle \lim_{n \to \infty}{ N_{p}(f_{n}-f)}=0$
thx
Hint (but give the full details): recall Fatou's lemma:
If $\{g_n\}$ is a sequence of non-negative measurable function, then $\int_E \liminf_{n\to+\infty}g_nd\mu\leq\liminf_{n\to+\infty}\int_E g_nd\mu.$
Apply it to $g_n:=|f_n|^p$ to get that $f\in\mathcal L^p$ and to $g_n=|f_n-f_m|^p$, for $m$ fixed an large enough, using the fact that $\{f_n\}$ is Cauchy in $\mathcal L^p$.