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Riemann Stieltjes Integral Sign Video

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Definition

Let P = {x_0, x_1, ... , x_n} be a partition of [a, b] and let t_k be a point in the subinterval [x_(k-1), x_k]. A sum of the form S(P, f, g) = sum(f(t_k)*(delta g_k)) is called a Riemann-Stieltjes sum of f with respect to a. We say f is Riemann-integrable with respect to g on [a, b], and we write "f is an element of R(g) on [a, b]" if there exists a number A having the following property: For every epsilon > 0, there exists a partition P_epsilon of {a, b} such that for every partition P finer than P_epsilon and for every choice of the points t_k in [x_(k-1), x_k], we have |S(P, f, a) - A| < epsilon.

Source: Mathematical Analysis, second edition by Tom M. Apostol

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+ Triangle Inequality *

+ Field Axioms *

+ Order Axioms *

+ Inductive Set *

+ Euclid's Lemma *

+ Completeness Axiom *

+ Approximation Property *

+ Archimedean Property *

+ Cauchy Schwarz Inequality *

+ Stereographic Projection *

+ Imaginary Unit *

+ Principal Argument *

+ Peano's Axioms *

+ Interior Point *

+ Component Interval *

+ Adherent Point *

+ Accumulation Point *

+ Derived Set *

+ Cantor Intersection Theorem *

+ Bounded Set *

+ Metric Space *

+ Square Metric *

+ Heine Borel Covering Theorem *

+ Compact Set *

+ Lindelof Covering Theorem *

+ Fixed Point Theorem *

+ Euclidean Metric *

+ Cauchy Condition *

+ Fixed Point *

+ Inverse Image *

+ Bolzano's Theorem *

+ Disconnectedness *

+ Arcwise Connectedness *

+ Bounded (Function)

+ Heine's Theorem *

+ Discontinuity *

+ Differentiable

+ Removable Discontinuity *

+ Jump Discontinuity *

+ Monotonic Function *

+ Rolle's Theorem *

+ Cauchy's Mean Value Theorem *

+ Taylor's Theorem *

+ Partial Derivatives *

+ Cauchy Riemann Equations *

+ Bounded Variation *

+ Measure Zero *

+ Total Variation *

+ Rectifiable Curve *

+ Non Rectifiable Curve *

+ Geometric Series *

- Riemann Stieltjes Integral *

+ Integration By Parts *

+ Step Function *

+ Euler's Summation Formula *

+ Riemann's Condition *

+ Oscillation *

+ Orientation Reversing *

+ Limit Superior *

+ Limit Inferior *

+ Telescoping Series *

+ Comparison Test *

+ Limit Comparison Test *

+ Dirichlet's Test *

+ Abel's Test *

+ Little Oh Notation *

+ Rearrangement *

+ Double Sequence *

+ Double Series *

+ Mertens' Theorem *

+ Dirichlet Product *

+ Cesaro Summability *

+ Infinite Product *

+ Uniform Boundedness *

+ Uniform Bound *

+ Almost Everywhere *

+ Weierstrass M Test *

+ Space Filling Curve *

+ Pointwise Convergence *

+ Bounded Convergence *

+ Big Oh Notation *

+ Mean Convergence *

+ Disk Of Convergence *

+ Series Expansion *

+ Taylor Series *

+ Binomial Coefficient *

+ Upper Function *

+ Lebesgue Integrable *

+ Interval Of Convergence *

+ Binomial Series *

+ Improper Riemann Integrable *

+ Cauchy Principle Value *

+ Orientation Preserving *

+ Continuously Differentiable *

+ Zeta Function *

+ Riemann Zeta Function *

+ Measurable Functions *

+ Characteristic Function *

+ Lebesgue Mesasure *

+ Gamma Function *

+ Dirichlet Integral *

+ Semimetric Space *

+ Riesz Fischer Theorem *

+ Fourier Series *

+ Square Integrable *

+ Orthogonal System *

+ Orthonormal System *

+ Dirichlet's Kernel *

+ Jordan's Test *

+ Fourier Integral Theorem *

+ Dini's Test *

+ Directional Derivative *

+ Integral Transform *

+ Riemann's Localization Theorem *

+ Total Derivative *

+ Gradient Vector *

+ Stationary Point *

+ Riemann Integrable *

+ Lagrange Multiplier *

+ Multiple Integral *

+ Triple Integral *

+ Saddle Point *

+ Functional Equation *

+ Quadratic Form *

+ Coordinate Transformation *

+ Contour Integral *

+ Jacobian Matrix *

+ Jacobian Determinant *

+ Double Integral *

+ Fubini's Theorem *

+ Beta Function *

+ Convolution *

+ Jordan Measure *

+ Open Mapping Theorem *

+ Inner Product *

+ Jordan Curve *

+ Piecewise Smooth *

+ Positively Oriented Circle *

+ Linear Homotopy *

+ Negatively Oriented Circle *

+ Liouville's Theorem *

+ Maximum Modulus Principle *

+ Minimum Modulus Principle *

+ Laurent Series *

+ Deleted Neighborhood *

+ Isolated Singularity *

+ Removable Singularity *

+ Gaussian Sum *

+ Conformal Mapping *

+ Mobius Transformation *

+ Essential Singularity *

+ Simple Pole *

+ Radius Of Convergence *

+ Complex Residue *

+ Monotonic Sequence *

+ Integral Test *

+ Implicit Function Theorem *

+ Bolzano Weierstrass Theorem

+ Mean Value Theorem

+ Cauchy Product *

+ Irremovable Discontinuity *

+ Complex Number *

+ Analytic Function *

+ Abel's Limit Theorem *

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