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 $\LL^p \left({\mu}\right)$.

Suppose that the pointwise limit $f := \displaystyle \lim_{n \mathop \to \infty} f_n$ 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.