Positivity 10 (2006), 737–753
2006 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040737-17, published online July 11, 2006
Some Measurability Results and Applications
to Spaces with Mixed Family-norm
Abstract. A space with mixed family-norm consists of all functions x on a
product space such that the function y(t)=x(t, ·)
belongs to V (here,
U(t)andV denote given K¨othe spaces). Conditions for the measurability
of y are given, and the K¨othe dual of such spaces is determined. For this
purpose a generalization of the Luxemburg-Gribanov theorem for ‘uniformly
measurable’ functions is proved. This result is also formulated for vector func-
Mathematics Subject Classiﬁcation (2000). 46E30; 28A20; 28A35.
Keywords. K¨othe space, ideal space, Banach function space, space with mixed
norm, space with mixed family-norm, measurable vector functions on product
spaces, Luxemburg-Gribanov theorem.
If U (t)(t ∈ T ) is a family of (semi-)normed function spaces over S,andV is another
function space over T , one may deﬁne a ‘family tensor product’ [U(·) → V ]asthe
space of all functions x on T × S such that x(t, ·) ∈ U (t)andy(t)=x(t, ·)
belongs to V . We are interested in the special case that T and S are measure
spaces and that U(t)andV are so-called K¨othe spaces (for example Lebesgue or
Orlicz spaces). The corresponding space [U(·) → V ] is equipped with the natu-
ral (semi-)norm x = y
. Such spaces arise naturally in the study of partial
k(t, s, σ)z(t, σ)dσ (t, s) ∈ T × S.
The author thanks Y. Abramovich and A. Martellotti for valuable comments and suggestions.
This paper was written in the framework of a DFG project (Az. AP 40/15-1). Financial support
by the DFG is gratefully acknowledged.