## Lecture Notes: Lebesgue Theory of Integration

Reference: Modern Real Analysis, Second Edition, by W. Ziemer (with Monica Torres).

Lecture 3: Additivity of outer measures.

Lecture 4: More properties of outer measures. Definition of sigma-algebras.

Lecture 5: Caratheodory outer measure.

Lecture 6: Introduction to Lebesgue measure.

Lecture 7: Approximation with open, closed, G-deltas and F-sigmas sets.

Lecture 9: Non-measurable sets.

Lecture 10: Lebesgue-Stieltjes measure. Hausdorff measure.

Lecture 11: Hausdorff outer measure is Caratheodory and Borel regular.

Lecture 12: Hausdorff dimension.

Lecture 14: Measurable functions.

Lecture 15: Characterization of measurable function.

Lecture 16: The Cantor-Lebesgue function.

Lecture 17: Composition of measurable function. Convergence almost everywhere.

Lecture 20: Definition and properties of the integral.

Lecture 21: More properties of the integral.

Lecture 22: Fatou's Lemma. Monotone Convergence Theorem.

Lecture 24: Improper integrals and Lebesgue integrals.

Lecture 25: Lp spaces. Holder inequality.

Lecture 27: Vitali's Convergence Theorem.

Lecture 28: Radon-Nikodym Theorem. Riesz Representation Theorem.

Lecture 29: Proof of Riesz Representation Theorem.

Lecture 31: Convolutions Cavalieri's Theorem.

Lecture 32: Vitali's Covering Theorem.

Lecture 34: Radon Measures. Derivatives of Radon Measures.

Lecture 35: Fundamental Theorem of Calculus. Part I.

Lecture 36: Functions of bounded variation.Absolutely continuous functions.

Lecture 38: Fundamental Theorem of Calculus part II.

Lecture 39: Functions of bounded variation and minimal surfaces.