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

Then the following are equivalent:


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

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

From $(1)$ to $(2)$
This follows from the reverse triangle inequality:
 * $\size {\norm f_p - \norm {f_n}_p} \le \norm {f - f_n}_p$

From $(2)$ to $(1)$
By Power of Absolute Value is Convex Real Function, we have:
 * $\forall a,b \in \R : \size {\frac {a-b} 2}^p \le {\size a}^p + {\size {-b} }^p = {\size a}^p + {\size b}^p$

In particular:
 * $\map {h_n} x := 2^p \paren {\size {\map f x}^p + \size {\map {f_n} x}^p} - \size {\map f x - \map {f_n} x}^p$

is a positive measurable function.

By Fatou's Lemma:
 * $\ds \int \liminf_{n \mathop \to \infty} h_n \rd \mu \le \liminf_{n \mathop \to \infty} \int h_n \rd \mu$

Observe:

and:

Thus it follows:
 * $\ds 2^{p+1} \norm f_p ^p \le 2^{p+1} \norm f_p ^p - \limsup_{n \mathop \to \infty} \int \size {f - f_n}^p \rd \mu$

By Real Number Ordering is Compatible with Addition, adding $- 2^{p+1} \norm f_p ^p$ to the both sides:
 * $\ds 0 \le - \limsup_{n \mathop \to \infty} \int \size {f - f_n}^p \rd \mu$

By Order of Real Numbers is Dual of Order of their Negatives:
 * $\ds \limsup_{n \mathop \to \infty} \int \size {f - f_n}^p \rd \mu \le 0$