Riesz's Convergence Theorem

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

Let $p \in \R$, $p \ge 1$.

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 $f := \displaystyle \lim_{n \mathop \to \infty} f_n$ exists $\mu$-almost everywhere, and that $f \in \map {\LL^p} \mu$.

Then the following are equivalent:


 * $(1): \quad \displaystyle \lim_{n \mathop \to \infty} \norm {f - f_n}_p = 0$
 * $(2): \quad \displaystyle \lim_{n \mathop \to \infty} \norm {f_n}_p = \norm f_p$

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