Measure and Integration Theory || Lebesgue Integral || Unit-3 || Lecture -1 || M.Sc Mathematics скачать в хорошем качестве

Measure and Integration Theory || Lebesgue Integral || Unit-3 || Lecture -1 || M.Sc Mathematics

Welcome to our lecture series on Measure and and Integration Theory! 🎓 This is the Third Unit of Measure and Integration Theory This comprehensive series is designed for students, researchers, and mathematics enthusiasts aiming to build a strong foundation in measure theory—a critical area of modern analysis with applications in probability, functional analysis, and beyond. Topics Covered: 📚 Measurable Functions: ⚫Definition and examples of measurable functions ⚫Properties of measurable functions ⚫Convergence concepts (pointwise, almost everywhere, etc.) ⚫Simple functions and their role in integration Approximation of a measurable Function by a sequence of simple functions Measurable Function as nearly continuous functions Egoroff Theorem Lusin Theorem Convergence in Measure and F. Riesz Theorem Almost Uniform Convergence Non Measurable Function Non Measurable sets Equivalent formulation of measurable sets in terms of open , closed F Sigma and G Sigma Sets Borel Sets algebra of measurable sets Lebesgue measure of a set of real Numbers 📚 Measure Theory: Introduction to sigma-algebras and measures Construction of measures (e.g., Lebesgue measure) Key theorems: Monotone Convergence, Dominated Convergence, Fatou’s Lemma Integration with respect to a measure Vitali Covering Lemma Differentiation of monotonic functions Function of bounded variation Differentiation of indefinite integral Fundamental Theorem of Calculus Absolutely continuous functions and their properties Lebesgue Convergence theorem General Lebesgue Integral Monotone Convergence theorem Bounded convergence Theorem Lebesgue Theorem Riemann Integrals function Integration of non negative functions Fatou Lemma Shortcomings of Riemann Integrals Lebesgue Integral of a bounded function over a set of finite measure and its properties Egoroff Theorem Lusin Theorem CONVERGENCE IN MEASURE 💻💻Each lecture includes clear explanations, intuitive examples, and detailed proofs to help you grasp both the theoretical and practical aspects of these concepts. 🙄Who is this for? 📝📕📚Undergraduate and graduate students Aspirants preparing for competitive exams (e.g., GRE Math, CSIR NET, GATE, PhD Entrance) 📚Researchers and professionals exploring advanced mathematics 📖 Don’t forget to like, subscribe, and hit the notification bell to stay updated on our latest uploads. Questions? Drop them in the comments below, and we’ll be happy to help! 🔗 For supplementary materials and lecture notes , Contact me on 9813155942 and you can Whatsapp me on this Number if you have any queries regarding lectures. Happy learning!😊🙏 #Set functions #Intuitive idea of measure #Elementary properties of measure, #Measurable sets and their fundamental #properties. #Lebesgue measure of a set of real numbers #Algebra of measurable sets #Borel set #Equivalent formulation of measurable sets in terms of open, Closed, F and G sets #Non measurable sets #Measurable functions and their equivalent formulations #Properties of measurable functions. #Approximation of a measurable function by a sequence of simple functions #Measurable functions as nearly continuous functions #Egoroff theorem #Lusin theorem #Convergence in measure and F. Riesz theorem #Almost uniform convergence #Shortcomings of Riemann Integral #Lebesgue Integral of a bounded function over a set of finite measure and its properties #Lebesgue integral as a generalization of Riemann integral #Bounded convergence theorem #Lebesgue theorem regarding points of discontinuities of Riemann integrable functions #Integral of non-negative functions #Fatou Lemma #Monotone convergence theorem #General Lebesgue Integral #Lebesgue convergence theorem #Vitali covering lemma #Differentiation of monotonic functions Function of bounded variation and its representation as difference of monotonic functions #Differentiation of indefinite integral Fundamental theorem of calculus #Absolutely continuous functions and their properties. #advancedcalculus #measurable #measureandintegrationtheory #measuretheory #integrationtheory #functionalanalysis #measuretheoryandintegrationtheory #mscmathematics #Mscmathematics Let me know if you'd like this tailored further!