Measurably entire functions and their growth

In 1997 B. Weiss introduced the notion of measurably entire functions and proved that they exist on every arbitrary free C-action defined on a standard probability space. In the same paper he asked about the minimal possible growth rate of such functions. In this work we show that for every arbitrary free C-action defined on a standard probability space there exists a measurably entire function whose growth rate does not exceed exp(exp[log^p |z|]) for any p > 3. This complements a recent result by Buhovski, Glücksam, Logunov and Sodin who showed that such functions cannot have a growth rate smaller than exp(exp[log^p |z|]) for any p < 2.