On vector measures

By J. Diestel, B. Faires. 1974

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In Pure Mathematics

Citations in the corpus

Minimax Estimation of Kernel Mean Embeddings (2016, background): “The condition in (43) ensures that ∫ ‖r(x)‖HdQ(x) ≤ √ Ck ∞ for any Q ∈ M1+(X ) and therefore ∫ r(x) dQ(x) is well defined as a Bochner integral for any Q ∈ M1+(X ) (Diestel and Uhl, 1977, Theorem 2, p.45).”
Kernel Mean Shrinkage Estimators (2014, background): “A kernel mean is deﬁned with respect to a probability distribution P over a measurable space X by where µ P is a Bochner integral (see, e.g. , Berlinet and Thomas-Agnan (2004) for a deﬁnition of Bochner integral) and H is an RKHS endowed with a reproducing kernel k : X × X → R that is measurable.”
Density Estimation in Infinite Dimensional Exponential Families (2013, background): “…operator as R h − ⊗ − i R k k ∞ x is p 0 -integrable in the Bochner sense (see [11, Deﬁnition 1, Theorem 2]), and therefore it follows from [11, Theorem 6] that J ( f ) = ( f − f 0 ) ⊗ ( f − f 0 ) , C x p 0 ( x ) dx HS , where C := C x p 0 ( x ) dx is the Bochner integral of C x , thereby…”
Measurable Schur Multipliers and Completely Bounded Multipliers of the Fourier Algebras (2002, background): “For our purposes it is sufficient to consider only weakly measurable such functions in the sense of [10].”
A coding of separable Banach spaces. Analytic and coanalytic families of Banach spaces (2002, methodology): “For definition and general results about RNP, we refer to [D-U], and for complete proofs of the statements below, to [Bos].”
Penalized triograms: total variation regularization for bivariate smoothing (2004, background): “Only for k = 1, and m arbitrary, does the classical univariate definition of Jordan (1881) adapt in a straightforward way; see Dinculeanu (1967).”
Multivalued evolution equations with nonlocal initial conditions in Banach spaces (2007, background): “In addition, K is uniformly integrable, hence, by [18], p.”
The Structure of Translation-Invariant Spaces on Locally Compact Abelian Groups (2015, unknown): “A standard reference for vector-valued integrals is Diestel and Uhl [11].”
Measurable selectors and set-valued Pettis integral in non-separable Banach spaces☆ (2009, background): “Our unexplained terminology can be found in our standard references for multi-functions [7, 27], Banach spaces [16] and vector integration [13, 42].”