Why sigma algebra and not other systems of sets? I got a short question. Why we are considering sigma-algebras and not other systems of sets in measure-theory? Why not dynkin-system for example?

The Dynkin-system is poor for the purpose of measure theory. In measure theory (probability theory) calculating the measure of a sets over which a sequence of functions converges is important.
Let, for instance, $\{f_n\}$ is a sequence of measurable functions. Then te set
$$\{x: \lim_{n \rightarrow \infty}f_n(x)=f(x)\}$$ cannot be expressed in general in a Dynkin system.
Why not? Because
$$\{x: \lim_{n \rightarrow \infty}f_n(x)=f(x)\}=\bigcap_{n=1}^{\infty}\bigcup_{N=1}^{\infty}\bigcap_{m=N}^{\infty}\{x:|f_m(x)-f(x)|\le\frac{1}{n}\}.$$ 
Such a set cannot be expressed in a Dynkin system unless it contains at least the finite intersections of its sets.
Saying that a Dynkin system is too special deserves a down vote!
