Introduction To Lebesgue Integration

In this article, we define the integral of real-valued functions on an arbitrary measure space and derive some of its basic properties. We refer to this integral as the Lebesgue integral, whether or not the domain of the functions is subset of equipped with Lebesgue measure. The Lebesgue integral applies to a much wider class of functions than the Riemann integral and is better behaved with respect to point wise convergence. We carry out the definition in three steps: first for positive simple functions, then for positive measurable functions, and finally for extended real-valued measurable functions and gives the proof of the Fatou’s Lemma and at the end proves the Lebasgue Dominated Convergence Theorem.