Measure theory studies measurable spaces and measure spaces.
Measure theory is the field of mathematics that grew out of the Lebesgue integral and Kolmogorov's axioms for probability.
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.
Probability theory studies special class of measures, so called probability measures which are normalized to unity.
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.
There is a generalization, the noncommutative measure theory, which is more or less the study of von Neumann algebra, see Connes (1995).
Ernst-Erich Doberkat, Measures and all that — A Tutorial, arxiv/1409.2662
Alain Connes (1995); Noncommutative Geometry.
D.H. Fremlin (2001); Measure Theory; 5 volumes, web.
Discussion via Boolean toposes is in