This course is designed to offer a basic introduction to measure theory and Lebesgue’s integral. The topics to be covered are: countable and uncountable sets, countability of the rationals,  uncountability of the reals, measurable sets and functions, the Lebesgue’s integral where E is a measurable subset of the real line and f is measurable on E, the  spaces as metric spaces, Cauchy sequences in  spaces, completeness of spaces, the Riesz-Fischer theorem and Mean convergence in the space .