The Bochner Integral

a₁ absolute convergence arbitrary assume axioms b₁ Banach space belong Bochner integrable function Bochner measurable bounded carrier bounded interval brick functions Chapter characteristic function consequently continuous functions convergence in measure converges absolutely converges almost everywhere Daniell integral defined definition denote disjoint elements equality equation equivalent exists a.e. f is integrable f(xo f₁ fact finite number fn→f follows Fubini theorem full derivatives function f ƒ and g given h₁ h₂ HEM-integral Hence holds implies inequality integrable sets K₁ Lebesgue integrable Lemma limit linear locally integrable functions matrix measurable functions measurable set modulus Moreover non-negative null function null set obtain open set positive numbers Proof properties proves R-integrable real numbers real valued functions satisfies sequence step function subsequence subset Theorem 3.1 upper integral vanishes write