Measure theory is the field of mathematics that grew out of the Lebesgue integral and Kolmogorov's axioms for probability?.
Some subfields and applications
The general measure theory studies general notions and constructions in measure theory, like the connection to integration, the measure spaces, derivation by measure, Caratheodory construction? and so on.
Measure theory is very much having a central role in studying so called ergodic theory of dynamical system.
Geometric measure theory is the geometric study of measures of subsets of Euclidean space and the measure theoretic aspects of various geometric objects, like the integration of classes of currents and their extremization properties.