Lebesgue Sets Immeasurable Existence

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 gures. 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 speci c area todaycalled the measure and integral theory. Essential contributions to building this theory was 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. There exists the sets which are not Lebesgue measurable and the sets which are Lebesgue measurable but are not Borel measurable. Hence B L P(X).