Real Analysis part 4 (Riemann Stieltjes Integral)

Name: Real Analysis part 4 (Riemann Stieltjes Integral)
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Riemann Stieltjes Integral, Riemann, Stieltjes integral, Riemann sums, Riemann, Integrals, Riemann integrals

Created byJaswinder Kaur

Last updated 10/2024

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What you'll learn

The Real Analysis is a very important and vast branch of Mathematics, applied in Higher Studies, and Riemann Stieltjes Integral is one of its content
This course serves as an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continous probability.
It also serves as an instructive and useful precursor of the Lebesgue Integral.
Students will learn that Riemann sums give us the systematic way to find the area of a curved surface when we know the mathematical function for that curve.

Course content

6 sections • 37 lectures • 8h 6m total length

Riemann Stieltjes Integral Introduction23:01
Introduce the Riemann-Stieltjes integral by defining partitions and mesh, upper and lower sums, refinements, and the common value of bounds as the integral.
Riemann Stieltjes Integral Continued....9:25
Explore the Riemann-Stieltjes integral on [a,b] for a monotone increasing alpha, bounded on interval, and construct upper and lower sums via partitions and delta alpha to determine when they coincide.
Riemann Stieltjes Integral Assignment 110:03
This assignment shows that for a real bounded f on [A,B] and increasing α, m(α_B−α_A) ≤ Lpα ≤ Upα ≤ M(α_B−α_A).
Riemann Stieltjes Integral Assignment 213:34
Refine partitions in the Riemann–Stieltjes integral, proving lp alpha ≤ lp star alpha and related bounds. Use p, p star, b, and b star to illustrate refinement effects on sums.
Riemann Stieltjes Integral Assignment 39:52
This lecture shows that for a real bounded function on [a,b] with α increasing, lp alpha ≤ lower Riemann–Stieltjes integral ≤ upper Riemann–Stieltjes integral ≤ up alpha.
Riemann Stieltjes Integral Assignment 422:03
Prove that for every epsilon, there exists a partition with the upper minus lower sums of the Riemann–Stieltjes integral less than epsilon, and that this holds for all refinements.
Riemann Stieltjes Integral Assignment 55:32
Show that for a bounded f on [a,b], the sum of |f(s_i)-f(t_i)| delta alpha_i is bounded by upf alpha minus lpf alpha, which is less than epsilon.
Riemann Stieltjes Integral Assignment 614:02
This lecture proves that a continuous f on [a,b], with alpha increasing, is Riemann-Stieltjes integrable on [a,b] using uniform continuity and epsilon-delta arguments.
Riemann Stieltjes Integral Assignment 7(Expected)7:02
Given f is monotonic on [a,b] and alpha is increasing and continuous, f is Riemann integrable on [a,b] with respect to alpha.
Riemann Stieltjes Integral Assignment 816:50
the lecture proves that a bounded f with finitely many discontinuities on [a,b], with alpha continuous at those points, is Riemann integrable, using partitions to bound upper and lower sums.
Riemann Stieltjes Integral Assignment 918:04
Demonstrates that if f is bounded and Riemann integrable on [a,b] and φ is continuous on [m,M], then h(x)=φ(f(x)) is Riemann integrable on [a,b], using uniform continuity and partition estimates.

Riemann Stieltjes Integral Assignment 1013:20
Explore the linearity, additivity, and comparison properties of the Riemann–Stieltjes integral on [a,b], including proofs when F1 and F2 are integrable, using partition sums S_P alpha and epsilon.
Riemann Stieltjes Integral Assignment 116:16
Demonstrate the linearity of the Riemann–Stieltjes integral by showing that c f is integrable and ∫_a^b c f dα = c ∫_a^b f dα, using partitions and an epsilon argument.
Riemann Stieltjes Integral Assignment 127:13
Demonstrate additivity of the Riemann–Stieltjes integral on adjacent intervals, showing ∫_a^b f dα = ∫_a^c f dα + ∫_c^b f dα when f is integrable on [a,c] and [c,b].
Riemann Stieltjes Integral Assignment 137:28
Demonstrates the additivity of the Riemann-Stieltjes integral for alpha1 and alpha2 on [a,b], using partition sums and an epsilon argument.

Riemann Stieltjes Integral Assignment 1413:34
If the limit of SPF_alpha sums exists as partition norm tends to zero, then f is Riemann-Stieltjes integrable on [a,b] with respect to alpha, and this limit equals alpha integral.
Riemann Stieltjes Integral Assignment 1512:50
If F1 and F2 are Riemann integrable on [a,b] and F1(x) <= F2(x), then the integral of F1 over [a,b] is no larger than that of F2. It covers subintervals.
Riemann Stieltjes Integral Assignment 1610:14
Explore the Riemann–Stieltjes integral on a closed interval [A,B] for an increasing alpha, proving limit convergence via uniform continuity, partitions, and epsilon–delta estimates.
Riemann Stieltjes Integral Assignment 1720:14
Shows that, for increasing alpha on [a,b], f is Riemann-Stieltjes integrable with respect to alpha iff f is integrable against alpha', and integrals equal ∫ f(x) alpha'(x) dx.
Riemann Stieltjes Integral Assignment 1811:32
Master the change of variables in the Riemann Stieltjes integral by defining beta from alpha and g from f, and proving integral equality under the transformation.

Integration and Differentiation Assignment 122:19
The lecture shows that F(x)=∫_a^x f(t) dt is continuous on [a,b]. It proves that if f is continuous at x0, then F is differentiable at x0 with F'(x0)=f(x0).
Fundamental Theorem of Calculus Assignment 27:58
Prove the fundamental theorem of calculus: if F is differentiable on [a,b] with F' = f, then ∫_a^b f = F(b) − F(a), using partitions and the mean value theorem.
Integration by parts Assignment 38:50
Derive the integration by parts formula for differentiable F and G on [a, b], showing ∫_a^b F G' = F(b)G(b) − F(a)G(a) − ∫_a^b F' G.
The use of Greatest Integer Function Assignment 412:24
Using the unit step function, prove that for a bounded f continuous at s, the Riemann–Stieltjes integral ∫_a^b f dα equals f(s), with α(x) = I_x − s.
Assignment 516:37
Demonstrates a Riemann–Stieltjes integral formula for a nonnegative step function α based on a convergent C_n series, proving ∫_A^B f dα = Σ C_n f(s_n).

Integration of Vector Values Functions15:03
Learn to integrate vector-valued functions with respect to an increasing alpha on [a,b] (Riemann–Stieltjes), where each component is Riemann integrable and the vector integral satisfies a Schwarz inequality.
Rectifiable Curves24:20
Define rectifiable curves as continuous mappings from [a, b] into an arc, with length lambda_gamma; show that if gamma' is continuous, gamma is rectified and lambda_gamma equals ∫_a^b |gamma'(t)| dt.

Riemann Steiltjes Integral Examples Assignment 112:14
Revisit the criteria for the Riemann integral on closed intervals using upper and lower sums, partitions, and epsilon bounds, including the irrational–rational example that is not integrable.
Riemann Steiltjes Integral Examples Assignment 210:55
Examine a bounded function on [a,b] defined by rationals and irrationals, show f is not Riemann integrable while f^2 is; discuss odd versus even powers and integrability.
Riemann Steiltjes Integral Examples Assignment 313:16
Compute lower and upper Riemann integrals for a function defined as x + x^2 on rationals and x^2 + x^3 on irrationals over [0,2], showing f is not Riemann integrable.
Riemann Steiltjes Integral Examples Assignment 48:12
Showcases calculating lower and upper integrals for a piecewise f on [0,1], with f equals sqrt(1-x^2) on rationals and 1-x on irrationals; prove not integrable since 1/2 ≠ pi/4.
Riemann Steiltjes Integral Examples Assignment 514:33
This lecture shows how Riemann sums converge to the integral on [a,b] using partitions, and applies it to the integral from 0 to 1 of f(x) dx and the case integral from 0 to 1 of 1/√(1+x^2) dx = ln(1+√2).
Riemann Steiltjes Integral Examples Assignment 614:05
the lecture proves that a continuous f on [a,b] with increasing alpha is Riemann-Stieltjes integrable, using uniform continuity, extreme values, and a delta-epsilon bound on sums.
Riemann Steiltjes Integral Examples Assignment 714:19
Explore the Riemann–Stieltjes framework for the zeta function, proving integral representations and convergence via integration by parts, breaking the domain, and using comparison tests with a convergent integral.
Riemann Steiltjes Integral Examples Assignment 810:30
This lecture derives two Riemann-Stieltjes identities from assignment 8 by converting sums to integrals and evaluating a log term and related x-integrals.
Riemann Steiltjes Integral Examples Assignment 918:30
Explore the Riemann-Stieltjes integral through assignment nine, showing quick evaluations via the conversion formula and integration by parts, and handling floor-function integrands across unit intervals.
Riemann Steiltjes Integral Examples Assignment 1010:25
Explore the Riemann–Stieltjes integral through assignment 10, proving the relation between a discrete sum and a Riemann–Stieltjes integral using summation formulas and integration by parts.

Requirements

The only pre-requisite is a knowledge of Calculus at high school level so that students of science and engineering who want deeper understanding of calculus or want to pursue subjects such as Theoretical Physics, Computational complexity, Statistics etc. would benefit from this course.

Description

Riemann Stieltjes Integral is a generalization of the Riemann Integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. It serves as an instructive and useful precursor of the Lebesgue Integral. and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability.

The definition of Riemann Stieltjes Integral uses a sequence of partitions P of the given interval [a,b].The integral, the, is defined to be the limit, as the norm (the length of the longest sub interval) of the partitions

The course introduces the definition and existence of the Riemann Stieltjes integral including the concepts of the following:

Common Refinement of a partition , Upper Riemann and Lower Riemann sum, Riemann Sum, Properties of Riemann Stieltjes Integral, Integration and Differentiation, The Fundamental Theorem Of Calculus,Integration Of Vector Valued Functions,

Rectifiable Curves,that also covers the Expected Theorems and Solved Examples based above contents.

This course also describes the various identities of Riemann Stieltjes integral.

Overall , every thing has been covered with lots of concepts and solved examples , assignments, exercises and almost all the expected theorems.

For any queries related to the course, I would be happy to assist you. Just ping me via Inbox. You will get a course completion certificate after finishing the course.

Who this course is for:

BSc. Graduates , Masters in Sciences( Mathematics), CSIR UGC NET and other Entrances Exams.

Real Analysis part 4 (Riemann Stieltjes Integral)

What you'll learn

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Course content

Introduction of Riemann Steiltjes Integral11 lectures • 2hr 29min

Properties of Riemann Stieltjes Integral4 lectures • 34min

Riemann Stieltjes Sum5 lectures • 1hr 8min

Riemann Stieltjes Integral Integration and Differentiation)5 lectures • 1hr 8min

Integration of Vector Valued Functions2 lectures • 39min

Riemann Steiltjes Integral Solved Examples10 lectures • 2hr 7min

Requirements

Description

Who this course is for: