Library / Integration / Chapter VI / § 2
Proposition 5
Tag 09AG, permanent. Cite it as [Bourbaki, Tag 09AG].
Integration, chapter VI, § 2. Vectorial measures . "
— Let q be a lower semi-continuous semi-norm on F. Let m be a q-majorizable measure, and let f be a function essentially integrable for m and such that ∫ f d m ∈ F \int f dm \in F ∫ f d m ∈ F
q ( ∫ f d m ) ≤ ∫ ∙ ∣ f ∣ d q ( m ) .
q\left( \int f dm \right) \leq \int^\bullet |f| dq(m).
q ( ∫ f d m ) ≤ ∫ ∙ ∣ f ∣ d q ( m ) . For,
q ( ∫ f d m ) = sup z ′ ∈ A q ′ ∣ ⟨ z ′ , ∫ f d m ⟩ ∣ ≤ sup z ′ ∈ A q ′ ∫ ∣ f ∣ d ∣ z ′ ∘ m ∣ ≤ ∫ ∙ ∣ f ∣ d q ( m )
q\left( \int f dm \right) = \sup_{z' \in A'_q} \left| \langle z', \int f dm \rangle \right| \leq \sup_{z' \in A'_q} \int |f| d|z' \circ m| \leq \int^\bullet |f| dq(m)
q ( ∫ f d m ) = z ′ ∈ A q ′ sup ⟨ z ′ , ∫ f d m ⟩ ≤ z ′ ∈ A q ′ sup ∫ ∣ f ∣ d ∣ z ′ ∘ m ∣ ≤ ∫ ∙ ∣ f ∣ d q ( m ) by virtue of (1) and the relation ∣ z ′ ∘ m ∣ ≤ q ( m ) |z' \circ m| \leq q(m) ∣ z ′ ∘ m ∣ ≤ q ( m ) z ′ ∈ A q ′ z' \in A'_q z ′ ∈ A q ′
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Transcribed from the printed volume. Not translated.