Riesz-Fischer Theorem

Theorem

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

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

The Lebesgue $p$-space $\map {\LL^p} \mu$, endowed with the $p$-norm $\norm {\cdot}_p$, is a Banach space.

Corollary

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

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

If a sequence $\sequence {f_k}$ in $\map {\LL^p} \mu$ converges to $f$,

then there is a subsequence $\sequence {f_{k_j}}$ that converges pointwise a.e. to $f$.

Proof

From Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach, to prove $\map {\LL^p} \mu$ is complete, it suffices to prove that every absolutely summable sequence in $\map {\LL^p} \mu$ is summable.

Let $\sequence {f_n}$ be an absolutely summable sequence in $\map {\LL^p} \mu$

Define:

$\ds \sum_{k \mathop = 1}^\infty \norm {f_k}_p =: B < \infty$

Also define:

$\ds G_n := \sum_{k \mathop = 1}^n \size {f_k}$

and:

$\ds G = \sum_{k \mathop = 1}^\infty \size {f_k}$

It is clear that the conditions of the Monotone Convergence Theorem (Measure Theory) hold, so that:

$\ds \int_X G^p = \lim_{n \mathop \to \infty} \int_X G_n^p$

By observing that:

we can also say that:

$\ds \int_X \size {G_n}^p \le B^p$

and therefore:

$\ds \lim_{n \mathop \to \infty} \int_X \size {G_n}^p \le B^p$

Therefore we have that:

$\ds \int_X G^p \le B^p < \infty$

This confirms:

$G \in \map {\LL^p} \mu$

In particular:

$G \in \map{\LL^p} \mu$

entails that:

$G < \infty$ a.e.

So $\sequence {f_k}$ is absolutely summable a.e..

By Absolutely Convergent Series is Convergent/Real Numbers:

$\ds F = \sum_{k \mathop = 1}^\infty f_k$

converges a.e.

Because $\size F \le G$:

$F \in \map {\LL^p} \mu$

It only remains to show that:

$\ds \sum_{k \mathop = 1}^n f_k \to F$ in $\norm {\cdot}_p$

which we can accomplish by Lebesgue's Dominated Convergence Theorem.

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Because $\ds \size {F - \sum_{k \mathop = 1}^n f_k}^p \le (2G)^p \in \map{\LL^1}\mu$, the theorem applies.

We infer:

$\ds \norm {F - \sum_{k \mathop = 1}^n f_k}_p^p = \int_X \size {F - \sum_{k \mathop = 1}^n f_k}^p \to 0$

Therefore by Definition of Lp Norm in $\map{\LL^p}\mu$ we have that $\ds \sum_{k \mathop = 1}^\infty f_k$ converges in $\map{\LL^p}\mu$.

This shows that $\sequence {f_k}$ is summable, as we were to prove.

$\blacksquare$

Source of Name

This entry was named for Frigyes Riesz and Ernst Sigismund Fischer.

Historical Note

The was proved jointly by Ernst Sigismund Fischer and Frigyes Riesz.

Fischer proved the result for $p = 2$, while Riesz (independently) proved it for all $p \ge 1$.

Sources

• 1999: Gerald B. Folland: Real Analysis: Modern Techniques and their Applications (2nd ed.): $6.1$ Theorem $6.6$
• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $12.7$