Functions of bounded variations , Absolute continuity and functions of bounded variation, Real Analysis , Vitali Cover
In this 4 hour 10 min Course on Functions of Bounded Variation of Real Analysis part 7 , the contents included are :
What you’ll learn
- Functions of bounded variations Detailed Concepts.
- Vitali’s Cover and its lemma , positive , negative and total variation.
- When the function is Indefinite Integral and its absolute continuity.
- Function is of bounded variation and its concept of Absolute Continuity.
Course Content
- Functions of Bounded Variation (Differentiation of Monotone Functions) –> 1 lecture • 9min.
- Functions of Bounded Variation –> 8 lectures • 2hr 3min.
- Absolute Continuity Lebesgue Measure Theory –> 3 lectures • 1hr 3min.
- MCQ Questions on Lebesgue Measure Theory –> 9 lectures • 56min.
Requirements
In this 4 hour 10 min Course on Functions of Bounded Variation of Real Analysis part 7 , the contents included are :
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.
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,
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.
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.