Let I be a Borel set of the real
line R, C the space of complex numbers, V a σalgebra of Borel subsets of I, μ a
fixed measure on V such that for any bounded set Q ∈ V , μ(Q) < ∞, g(λ,p) a
nonvanishing complex valued function μmeasurable in λ ∈ I such that g(λ,p)↑ in p
where p belongs to a fixed open interval (a,b), and S the set of μmeasurable
functions u from I into C such that u(λ)g(λ,p)≦ m for some p depending on
u,p ∈ (a,b), m ≧ 0 and m depending on u, and for all λ ∈ I. The purpose of this
paper is to prove the following:
Theorem 1. Let c(λ,δ) be a μ × μmeasurable function on I × I. For every
function u ∈ S the function
is well defined and y ∈ S if and only if for every p ∈ (a,b) there exists a q ∈ (a,b)
such that
for all (λ,δ) ∈ I × I and some m ≧ 0.
