Franks John. Notes on Measure and Integration

Arxiv.org, 2009. — 118 p.This text grew out of notes I have used in teaching a one quarter course on integration at the advanced undergraduate level. My intent is to introduce the Lebesgue integral in a quick, and hopefully painless, way and then go on to investigate the standard convergence theorems and a brief introduction to the Hilbert space of L2 functions on the interval.
The actual construction of Lebesgue measure and proofs of its key properties are relegated to an appendix. Instead the text introduces Lebesgue measure as a generalization of the concept of length and motivates its key properties: monotonicity, countable additivity, and translation invariance.Background and FoundationsThe Completeness of R
Sequencesin R
Set Theory and Countability
Open and Closed Sets
Compact Subsets of R
Continuousand Differentiable Functions
Real Vector SpacesThe Regulatedand Riemann IntegralsBasic Properties o fan Integral
Step Functionsand the Regulated Integral
The Fundamental Theorem of Calculus
The Riemann IntegralLebesgue MeasureNull Sets
Sigmaalgebras
Lebesgue Measure
The Lebesgue Density Theorem
Lebesgue Measurable Sets–SummaryThe Lebesgue Integral
Measurable Functions
The Lebesgue Integral of Bounded Functions
The Bounded Convergence TheoremThe Integral of Unbounded Functions
Non-negative Functions
Convergence Theorems
Other Measures
General Measurable FunctionsThe Hilbert Space
Square Integrable Functions
ConvergenceinL
Hilbert Space
Fourier SeriesLebesgue MeasureOuter Measure
Lebesgue Measurable Sets
A Non-measurable Set