Real Analysis part 7 Lebesgue Measure & Bounded varia

About this course

In this 4 hour 10 min Course on Functions of Bounded Variation of Real Analysis part 7 , the contents included are :1) Differentiation of monotone functionsDefinition of Vitali's Cover, Lemma on Vitali covering theoremThe definition of derivatives of function f at xIf f is an increasing real valued function on the interval [a,b] then f is differentiable almost everywhere.2) Functions of bounded variation:Positive variation, Negative variation, total variationJordan's theorem: A function f is of bounded variation on closed interval a and b if and only if f is the difference of two monotone real valued functions on closed interval a and b.If f is of bounded variation on closed interval a and b then derivative of f at x exists for almost all x in closed interval a and b.The conditions for which f is a continuous function on closed interval a and b then there exist c belonging to that interval,3) Differentiation of an integral:In this section we shall show that the derivative of indefinite integral of an integrable function is equal to the integrand almost everywhere. We begin by establishing some lemmas.Very Important : Lemma on integrable function on closed interval then function if defined is a continuous function of bounded variation on closed interval.The lemma on integrable function on closed interval and inetgration of f is 0 with limits x and t for all x belonging to closed interval then f(t) = 0 almost everywhere in closed interval a and b.The Lemma on bounded and measurable functions.The Theorems on integrable functions.4) Absolute Continuity:Definition of Absolutely continous.Lemma on Absolutely continuous on closed interval.Corollary on absolutely continuous.Lemma on Absolute continuous on closed interval and derivative of f is 0.Definit