The aim of this paper is to address the basic concepts that bind to measure theory. For this purpose, we have defined the general measure and the basic properties of the measure. In doing so, we were able to define the external and Lebesgue measures that were later used to explain measurable functions and properties associated with them. At the end of Chapter two, we have defined simple functions that are very important for integrable functions. To define integrable functions, we first define integrals of nonnegative functions and integrals of measurable func tions, and then integrable functions. After that we also proved and proved some important theorems such as ”Levi’s Monotone Convergence Theorem” or ”Lebesgue Domination Convergence Theorem”. We ended this thesis by connecting Riemann’s and Lebesgue’s integrals because the Lebesgue integral is defined for a much wider class of functions than the class of bounded functions.