Real Analysis part 7 Lebesgue Measure & Bounded varia

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 functions

Definition of Vitali's Cover, Lemma on Vitali covering theorem

The definition of derivatives of function f at x

If 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 variation

Jordan'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.

Definition of indefinite integral.

Theorem on indefinite integral.

Corollary on absolute continuous function.

Including detailed concept and theorems used.