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.

Proof

From $(1)$ to $(2)$

This follows from the reverse triangle inequality:

$\size {\norm f_p - \norm {f_n}_p} \le \norm {f - f_n}_p$

$\Box$

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:

\(\ds \int \liminf_{n \mathop \to \infty} h_n \rd \mu\)

\(=\)

\(\ds \int \liminf_{n \mathop \to \infty} \paren {2^p \paren {\size {\map f x}^p + \size {\map {f_n} x}^p} - \size {\map f x - \map {f_n} x}^p} \rd \mu\)

\(\ds \)

\(=\)

\(\ds \int 2^p \paren {\size {\map f x}^p + \size {\map f x}^p} - \size {\map f x - \map f x}^p \rd \mu\)

as $\ds f = \lim_{n \mathop \to \infty} f_n$ $\mu$-a.e.

\(\ds \)

\(=\)

\(\ds 2^{p+1} \norm {f_n}_p ^p\)

and:

\(\ds \liminf_{n \mathop \to \infty} \int h_n \rd \mu\)

\(=\)

\(\ds \liminf_{n \mathop \to \infty} \paren {2^p \int \size f^p \rd \mu + 2^p \int \size {f_n}^p \rd \mu + \int - \size {f - f_n}^p \rd \mu}\)

\(\ds \)

\(=\)

\(\ds \liminf_{n \mathop \to \infty} \paren {2^p \norm f_p ^p + 2^p \norm {f_n}_p ^p + \int - \size {f - f_n}^p \rd \mu}\)

\(\ds \)

\(=\)

\(\ds 2^p \norm f_p ^p + 2^p \lim_{n \mathop \to \infty} \norm {f_n}_p ^p + \liminf_{n \mathop \to \infty} \int - \size {f - f_n}^p \rd \mu\)

as $\ds \lim_{n \mathop \to \infty} \norm {f_n}_p$ exists

\(\ds \)

\(=\)

\(\ds 2^{p+1} \norm f_p ^p + \liminf_{n \mathop \to \infty} \int - \size {f - f_n}^p \rd \mu\)

as $\ds \lim_{n \mathop \to \infty} \norm {f_n}_p = \norm f_p$

\(\ds \)

\(=\)

\(\ds 2^{p+1} \norm f_p ^p - \limsup_{n \mathop \to \infty} \int \size {f - f_n}^p \rd \mu\)

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:

$\ds \limsup_{n \mathop \to \infty} \int \size {f - f_n}^p \rd \mu \le 0$

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Source of Name

This entry was named for Frigyes Riesz.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $12.10$