Theorem 5
Tag 08VI, permanent. Cite it as [Bourbaki, Tag 08VI].
Integration, chapter IV, § 5. Measurable functions and sets
— In order that a mapping f of X into a Banach space F be p-th power integrable (), it is necessary and sufficient that f be measurable and that be finite.
The condition is necessary: for, if then there exists a sequence of continuous functions with compact support that converges almost everywhere to f (, No. 4, Cor. 2 of Th. 3); by Th. 2 of No. 4, f is measurable.
To prove that the conditions are sufficient, we first establish a lemma:
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
Cited by
- § 5. Measurable functions and sets
- Corollary 3, § 5
- Corollary 4, § 5
- Proposition 20, § 5
- Remark, § 6
- Proposition 1 (Inequality of the mean), § 6
- Theorem 2 (Hölder’s inequality), § 6
- Corollary 1, § 6
- Corollary 5, § 6
- Lemma, § 1
- Corollary, § 3
- § 5. Measures defined by numerical densities
- Proposition 1, § 5
- Remark, § 6
- Scholium, § 8
- Corollary 2, § 8
- Corollary, § 2
- Proposition 10, § 2