Riesz's Convergence Theorem

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

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

Let $\left({f_n}\right)_{n \in \N}, f_n: X \to \R$ be a sequence in Lebesgue $p$-space $\mathcal{L}^p \left({\mu}\right)$.

Suppose that the pointwise limit $f := \displaystyle \lim_{n \to \infty} f_n$ exists $\mu$-almost everywhere, and that $f \in \mathcal{L}^p \left({\mu}\right)$.

Then the following are equivalent:


 * $(1): \quad \displaystyle \lim_{n \to \infty} \left\Vert{f - f_n}\right\Vert_p = 0$
 * $(2): \quad \displaystyle \lim_{n \to \infty} \left\Vert{f_n}\right\Vert_p = \left\Vert{f}\right\Vert_p$

where $\left\Vert{\cdot}\right\Vert_p$ denotes the $p$-seminorm.