Specker's theorem for Nöbeling's group

Specker proved that the group $\mathbb{Z} ^{\aleph_0}$ of integer-valued sequences is far from free; all its homomorphisms to $\mathbb{Z}$ factor through finite subproducts. Nöbeling proved that the subgroup $\mathcal{B}$ consisting of the bounded sequences is free and therefore has many homomorphisms to $\mathbb{Z}$. We prove that all ``reasonable'' homomorphisms $\mathcal{B}\to\mathbb{Z}$ factor through finite subproducts. Among the reasonable homomorphisms are all those that are Borel with respect to a natural topology on $\mathcal{B}$. In the absence of the axiom of choice, it is consistent that all homomorphisms are reasonable and therefore that Specker's theorem applies to $\mathcal{B}$as well as to $\mathbb{Z} ^{\aleph_0}$.