On soft mappings of the unit ball of Borel measures

The main result of this paper is two theorems. One of them asserts that the functor $U_\tau$ takes the 0-soft mappings between spaces of weight ${\leq}\,\omega_1$ and Polish spaces to soft mappings. The other theorem, which is a corollary to the first one, asserts that the functor $U_\tau$ takes the $\mathrm{AE}(0)$-spaces of weight ${\leq}\,\omega_1$ to $\mathrm{AE}$-spaces. These theorems are proved under Martin's axiom $\textup{MA}(\omega_1)$. The results cannot be extended to spaces of weight ${\geq}\,\omega_2$. For spaces of weight $\omega_1$, these results cannot be obtained without additional set-theoretic assumptions. Thus, the question as to whether the space $U_\tau(\mathbb R^{\omega_1})$ is an absolute extensor cannot be answered in ZFC. The main result cannot be transferred to the functor $U_R$ of the unit ball of Radon measures. Indeed, the space $U_R(\mathbb R^{\omega_1})$ is not real-compact and, therefore, $U_R(\mathbb R^{\omega_1})\notin\mathrm{AE}(0)$.