  The main purpose of this article is to provide new results on algebraic
independence of values of Mahler functions and their generalizations.
Simultaneously, we establish new measures of algebraic independence for these
values. Among the other things, we provide a measure of algebraic independence
for values of Mahler's functions at complex transcendental points, a result of
type which has never appeared in the literature in the past.
  As an example of application of our new measures of algebraic independence,
we prove that a Mahler number does not belong to the class $U$ in Mahler's
classification. Also, our results imply new examples, for $n\geq 1$ arbitrarily
large, of sets $\left(\theta_1,\dots,\theta_n\right)\in\mathbb{R}^n$ normal in
the sense of G.~Chudnovsky (1980).
