Math 541: Real Analysis

Real Analysis.

3

Prerequisite

Math 315, 343; 214 or 316.

Description

Rigorous treatment of differentiation and integration theory; Lebesque measure; Banach spaces.

Desired Learning Outcomes

Math 541 is currently the first half of a two-semester sequence on Lebesgue integration in Euclidean space and several related topics, but it is proposed that it become a one-semester course specifically on Lebesgue integration in Euclidean space and Fourier transforms.

Minimal learning outcomes

1. Lebesgue measure on Rn
• Inner and outer measures
• Construction of Lebesgue measure
• Properties of Lebesgue measure
• Effect of basic set operations
• Limiting properties
• Its domain
• Approximation properties
• Sets of outer measure zero
• Invariance w.r.t. isometries
• Effect of dilations
• Existence of nonmeasurable sets
2. Lebesgue integration on Rn
• Measurable functions
• Simple functions
• Approximation of measurable functions with simple functions
• The extended reals
• Integrating nonnegative functions
• Integrating absolutely-integrable functions
• Integrating on measurable sets
• Basic properties of the Lebesgue integral
• Linearity
• Monotonicity
• Effects of sets of measure zero
• Absolute continuty of integration
• Fatou's Lemma
• Monotone Convergence Theorem
• Dominated Convergence Theorem
• Differentiation w.r.t. a parameter
• Linear changes of variable
• Compatibility with Riemann integration
3. Fubini's Theorem for Rn
4. L1, L2, and L
• Completeness
• Approximation by smooth functions
• Continuity of translation
5. Fourier transform on Rn
• Convolutions
• Basic properties of Fourier transforms
• Composition with translation, dilation, inversion, differentiation, convolution, etc.
• Regularity of transformed functions
• Riemann-Lebesgue Lemma
• Inversion Theorem for L1
• Schwartz class
• Fourier-Plancherel Transform on L2
• Its inversion
• Isomorphism
• Fourier series
• Dirichlet and Fejér kernels
• L2 convergence
• Pointwise convergence
• Convergence of Cesàro means