A New Class of Banach Spaces with the Lebesgue Property

There are two main contributions in this paper. First, the \(X_{iw}\)-sum of Banach spaces is defined, following the work of Argyros and Motakis in [4]. In reality, this Banach space is an example of a \(U\)-sum of Banach spaces (e.g. in [12]), however, the alternate construction that is provided in this paper allows for a self-contained proof that it has the Lebesgue property if each summand has the Lebesgue property. This second result not only obtains a large new class of Banach spaces with the Lebesgue property, but it also shows that the non-asymptotic-\(\ell_{1}\) sum of infinite-dimensional Banach spaces still can have the Lebesgue property, which had not been established previously.