Let $f: X \to [0, \infty) \subset \mathbb R$ measurable where $X$ is a measure space. Let $f_n : X \to [0, \infty) $ be simple functions (i.e. linear combinations of characteristic functions of measurable sets) such that for each $x \in X$, $f_n(x) \leq f_{n+1}(x)$ and $f_n(x)$ converges to $f(x)$.
How can I prove that $ \|f_n - f \|_p = \left ( \int_X |f - f_n|^p d \mu\right )^{1/p} \xrightarrow{n \to \infty} 0$
I don't think this is right but if for $n > N_x$, $|f_n(x) - f(x)| \leq \varepsilon$, we can let $N = \sup_{x \in X} N_x$ to get $\|f_n - f\|_\infty \leq \varepsilon$ and then $ \|f_n - f \|_p = \left ( \int_X \|f - f_n\|^p d \mu\right )^{1/p} \leq \left ( \int_X \varepsilon^p d \mu\right )^{1/p} = \mu(X)^{1/p} \varepsilon $
But $\mu(X)$ could be infinite so I'm not sure what to do. Thanks.
Edit What assumptions do I need to make this true?