Strongly measurable functions and multipliers of $${\mathcal {M}}-$$ integrable functions
Tóm tắt
We investigate the integrability of Banach space valued strongly measurable functions defined on [0, 1]. In the case of functions given by
$$f=\sum _{n=1}^{\infty }x_n \chi _{E_n}$$
where
$$x_n\in X$$
and the sets
$$E_n$$
are Lebesgue measurable and pairwise disjoint subsets of [0, 1], there are well known characterizations for Bochner and McShane integrability of f. The absolute (resp. unconditional) convergence of
$$\sum _{n=1}^{\infty }x_n m(E_n)$$
is equivalent to Bochner (resp. McShane) integrability of f. We give some conditions for scalar McShane and weakly McShane integrability of f and their relation with unconditionally converging operators. We also study the space of vector valued multipliers of strongly McShane
$$(\mathcal {SM})-$$
integrable functions. We prove that if X is a commutative Banach algebra, with identity e of norm one, satisfying Radon–Nikodym property and
$$M: \mathcal {SM}\rightarrow \mathcal {SM}$$
is a bounded linear operator, then there exists
$$g\in L^\infty ([0,1],X)$$
such that
$$M(f)= fg$$
for all
$$f\in \mathcal {SM}$$
. Some results on multipliers of McShane
$$({\mathcal {M}})-$$
integrable functions are also derived.
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