Difference between revisions of "Math 541: Real Analysis"

Revision as of 10:42, 13 August 2008

Real Analysis.

3

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

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+# Lebesgue measure on <b>R</b><sup>n</sup>
+#* 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
+# Lebesgue integration on <b>R</b><sup>n</sup>
+#* 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
+# Fubini's Theorem for <b>R</b><sup>n</sup>
+# L<sup>1</sup>, L<sup>2</sup>, and L<sup>&#8734;</sup>
+#* Their completeness
+#* Approximation by smooth functions
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