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Henri Lebesgue, Emile Borel, Maurice Fréchet and Johann Radon are very important scientists that contribute to the development of the measure theory. Probability theory and ergodic theory can be seen as the principal applications of measure theory and Lebesgue integration. By using Lebesgue integration, one can define integrals on more general subsets of R or R. Similarly to that, in a different space, when one can specify the measure, then the integral on more general subsets of this space can also be defined. Moreover, if Riemann and Lebesgue integrations are considered, the difference can be seen more easily. As the example of the measure theory for the other spaces the probability theory and ergodic theory can be taken into account. The measure of the whole set in measure theory is taken as 1 and the events are considered as the measurable sets. Measures that are invariant under a dynamical system is the subject of ergodic theory.

HARDY-SOBOLEV-MAZYA INEQUALITY FOR NABLA TIME SCALE CALCULUS