I am wanting to prove that:
If $f_n$ converges to $f$ pointwise on a probability space $(\Omega,\mathscr{F},\mathbb{P})$ and $\left\vert f_n\right\vert \le g$ for some $g\in \mathscr{L}^p\left( \Omega,\mathscr{F},\mathbb{P}\right)$, where $p \ge 1$. Then $\lim_{n\rightarrow\infty} \left\Vert f - f_n\right\Vert_p =0$, or in other words, $f_n$ converges to $f$ in the $\mathscr{L}^p$-norm.
This question was asked here: Pointwise convergence implies $L^p$
But I don't understand how to apply Lebesgue's Dominated Convergence Theorem to get the result as they mentioned in the second answer. Also I am not too sure how the following is true:
\begin{equation} \int \left\vert f_n - f\right\vert^pd\mathbb{P} \le \int\left\vert f_n\right\vert^pd\mathbb{P} + \int\left\vert f\right\vert^pd\mathbb{P} \end{equation}
I know that we can apply Minkowski's inequality to obtain:
\begin{equation} \left(\int \left\vert f_n - f\right\vert^pd\mathbb{P} \right)^\frac{1}{p} = \left\Vert f_n - f\right\Vert_p \le \left\Vert f_n\right\Vert_p + \left\Vert f\right\Vert_p = \left(\int\left\vert f_n\right\vert^pd\mathbb{P}\right)^\frac{1}{p} + \left(\int\left\vert f\right\vert^pd\mathbb{P}\right)^\frac{1}{p} \end{equation}
I also have seen on Wikipedia, that I am to apply the Dominated Convergence to the function $h_n:=\left\vert f_n - f\right\vert^p$ which is dominated by $(2g)^p$, but I am not too sure how $h_n$ is dominated by $(2g)^p$.
Thanks in advanced,
We aim to apply the dominated convergence theorem to the sequence $(|f_n-f|^p)_n$. Clearly $|f_n-f|^p\to 0$, so that it remains to find the dominating function. Then $|f_n|\leq g$ implies $|f|\leq g$ a.e., so that $$ |f_n-f|^p \leq (|f_n|+|f|)^p \leq (2g)^p = 2^pg^p \in L^1 $$ holds a.e. Now the dominated convergence theorem yields $\int |f_n-f|^p \to 0$ and consequently $\|f_n-f\|_p\to 0$.