I want to prove the $L^p$ Dominated Convergence Theorem which says :
Let $\{ f_n \}$ be a sequence of measurable functions that converges pointwise a.e. on E to $f$. For$ 1 \leq p < \infty$, suppose that there is a function $g$ in $L^p(E)$ such that for all $n$, $|f_n|<g$ a.e. on $E$. Prove that $\{ f_n \} \to f$ in $L^p(E)$.
This is my attempt :
$$ |f_n-f|<|g|+|f| \ \ since \ |f_n – f|\leq |f_n| + |f| \leq |f|+|g| \leq 2 max\{|f|,|g|\} \Rightarrow |f_n – f|^p<2^p(|g|^p + |f|^p) $$
If I show that $2^p(|g|^p + |f|^p)$ is integrable over $E$ then the result follows by Lebesgue's Dominated Convergence theorem. But this is where I am having a question : If $\{ f_n \}$ is a sequence in $L^p(E)$ and $\{ f_n \} \to f$ pointwise a.e. on $E$, does this necessarily mean that $f \in L^p(E) $ ? If yes, the result follows after using $2^p(|g|^p + |f|^p)$ as the dominating function in LDC..
ADDED LATER
For the bold-faced question, I tried like this $$|f|=|f-f_n+f_n|\leq |f-f_n|+|f_n| \Rightarrow \int_E |f|^p \leq \int_E|f-f_n|^p+\int_E|f_n|^p$$ Is there a theorem that says if $\{f_n\}→f$ pointwise a.e. on E, $\{f_n\}→f$ uniformly on $E\setminus E_0$ where $m(E_0)$ ? If thats true, for any $\epsilon$ and for some $N$ , we can say $|f-f_N|^p < \epsilon^{p}$ and since $f_n \in L^p(E)$, this ensures that the $\int_E |f|^p < \infty$.
I'd appreciate if you tell me what is going wrong in my statement. Thank you
Best Answer
I think I have simpler proof.
If we know that $ f_n \to f $ in measure this means that there is subsequence $f_{n_k}$ converging to $f$ point wise a.e. We still have $\left| f_{n_k} \right| < g $ so the Dominated Convergence (for $L_1$) implies that $$ \intop \left| f_{n_k} - f \right|d\mu \to 0 $$
We know (form the point-wise converges) that $\left| f_{n_k} - f \right| \to 0 $ a.e so we can assume that $\left| f_{n_k} - f \right| < 1 $ a.e. This implies us
$$ 0\leq \intop \left| f_{n_k} - f \right| ^ p d\mu \leq \intop \left| f_{n_k} - f \right|d\mu \to 0 $$ Thus $$ \left\Vert f_{n_k} -f \right\Vert _p \to 0 $$
If we won't have $ \left\Vert f_{n} -f \right\Vert _p \to 0 $ (i.e. converges in $L^p$) we could construct subsequence of $f_n$ $h_i = f_{n_i}$ such that
$$ \left\Vert h_{i} -f \right\Vert _p > \epsilon $$ for some $\epsilon > 0 $. We would still have $\left\vert h_i \right\vert < g$ and $h_i \to f$ is measure, so we may construct subsequence $h_{i_k}$ (as done before) of $h_i$ such that $$ \left\Vert h_{i_k} - f \right\Vert _p \to 0 $$
which contradict the condition that defined $h_i$.