Lebesgue-stieltjes Measure on R

It is well known that the notion of measure and integral were released early enough in close connection with practical problems of measuring of geometric figures. Notion of measure was outlined in the early 20th century through H. Lebesgue's research, founder of the modern theory of measure and integral. It was developed concurrently a technique of integration of functions. Gradually it was formed a specific area today called the measure and integral theory. Essential contributions to building this theory were made by a large number of mathematicians: C. Carathodory, J. Radon, O. Nikodym, S. Bochner, J. Pettis, P. Halmos and many others. In the following we present several abstract sets, classes of sets, positive measure on a cr-algebra, Lebesgue measure on R and the particularly measure Lebesgue-Stieltjes measure on R.