A Result on Bounded Measurable Functions in Lp

Abstract

In this article we presented some of the results with which we dealt while doing the
Course Concluding Work on Measure and Integration Theory. Lebesgue integrable functions are functions that lie in a space called L^p Space, with p ∈ [1, ∞). At first we defined such space and, as one of the important results about it, we showed that it is a vector space. Then, after defining a norm for that space, we demonstrated that it is a normed vector space. With this purpose, we used three important inequalities: Young’s Inequality, Holder’s Inequality and Minkowsky’s Inequality. Afterwards we defined a distance through that norm and demonstrated that L^P space with that distance is a complete metric space. An Lebesgue Integrable Function must be a simple function, or otherwise there must be a simple function having properties similar to those of the function to be integrated. At the end of this work we presented a theorem which guarantees the existence of a simple function having properties similar to those of a function lying in L^P Space. Finally, we have obtained an application clarifying that the set of limited measurable functions is dense in the L^p space.