Approximation of functions by neural networks

We study the approximation of measurable functions on the hypercube by functions arising from affine neural networks. Our main achievement is an approximation of any measurable function $f \colon W_n \to [-1,1]$ up to a prescribed precision $\varepsilon>0$ by a bounded number of neurons, depending only on $\varepsilon$ and not on the function $f$ or $n \in \mathbb N$.