🧱 Real Analysis

"A measure is more than just a size; it's the foundation upon which we build the entire house of modern analysis."

📊 Course Metadata

• Department: School of Mathematical Sciences
• Prerequisites: Mathematical Analysis, Higher Algebra, Functions of a Real Variable, Point-Set Topology (Recommended)
• Course Nature: 48 Credit Hours, Graduate Course

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📚 References

This course does not use PPTs or other presentation tools; it is taught entirely via handwritten blackboard notes by the professor.

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🗺️ Syllabus & Navigation

Part I: Measures and Measure Spaces

• 1.1 Fundamentals of \(\sigma\)-algebras and Measure Spaces

• 1.2 Outer Measures and Carathéodory Extension

• 1.3 Lebesgue-Stieltjes Measures and Lebesgue Measures

Part II: Measurable Functions

• 2.1 Concepts and Properties of Measurable Functions

• 2.2 Limits and Decompositions of Measurable Function Sequences

Part III: Abstract Integration

• 3.1 Construction of Abstract Integrals and Simple Functions

• 3.2 The Three Convergence Theorems and Convergence in Measure

• 3.3 Product Measures and Fubini's Theorem

Part IV: Signed Measures and Differentiation

• 4.1 Signed Measures and Hahn-Jordan Decomposition

• 4.2 Absolute Continuity and the Radon-Nikodym Theorem

• 4.3 Differentiation on \(\mathbb{R}\) and the Fundamental Theorem of Calculus

Part V: \(L^p\) Spaces and Functional Foundations

• 5.1 \(L^p\) Spaces, Inequalities, and Completeness

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📝 Notes & Formatting Details

Construction Complete

This page is now substantially complete. Long theorems and exercise solutions in the notes feature a folded box design. If you spot any typos, please feel free to raise an Issue or contact me directly for corrections!

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