Pointwise Convergent Bounded Sequence in Lebesgue Space Converges in Norm

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p \in \R_{\ge 1}$.

Let $\sequence {f_n}_{n \mathop \in \N}, f_n: X \to \R$ be a sequence in Lebesgue $p$-space $\map {\LL^p} \mu$.

Suppose that the pointwise limit $f := \ds \lim_{n \mathop \to \infty} f_n$ exists $\mu$-almost everywhere.

Suppose that for some $g \in \map {\LL^p} \mu$, the pointwise inequality $\size {f_n} \le g$ holds for all $n \in \N$.

Then $f \in \map {\LL^p} \mu$, and:


 * $\ds \lim_{n \mathop \to \infty} \norm {f - f_n}_p = 0$

where $\norm {\, \cdot \,}_p$ denotes the $p$-seminorm.

Proof
Since:
 * $\ds \size f = \lim _{n \to \infty} \size {f_n} \le g$

$\mu$-almost everywhere, we have:
 * $\int \size f^p d \mu \le \int g^p d \mu < + \infty$

Thus:
 * $f \in \map {\LL^p} \mu$

Furthermore, since:
 * $\size {f_n - f} \le \size {f_n} + \size f \le 2 \size g$

we have:
 * $\size {f_n - f}^p \le 2^p \size g^p$

Since $2^p \size g^p$ is $\mu$-integrable: