Description  Date 
Location and Time 
Remarks 
Test 1  Oct 13  In class  
Final Exam  takehome 
Date  Material covered  HW problems and assignments  Remarks 
Aug 27  Basics. 
HW1 
This hw is optional; it is just to warmup. 
Sept 1  A bit of topology; compactness.  HW2  HW due Sept 8 
Sept 3  Topological vector spaces.  
Sept 8  More TVS, convexity.  HW3  HW due Sept 15 
Sept 10  Convex inequalities.  
Sept 15  Some functions spaces. Measures.  HW4  HW due Sept 22 
Sept 17  Properties of measures. Characterization in terms of inequalities.  
Sept 22  Some convergence results. Real, positive, and relatively bounded measures. Ordering.  
Sept 24  Student presentation: constructive definition of inductive topologies. Inequalities with positive measures. Sup and inf of measures.  HW5  
Sept 29  Operations with measures. Uniform convergence and uniform approximation. Lower semicontinuous functions.  
Oct 1  Upper integral and outer measure. Lebesgue outer measure.  HW6  Theorem proven in
class 
Oct 6 
More on upper integral and outer measure.  
Oct 8  Student presentations: properties of lower semicontinuous functions; uniform approximation; analytic HahnBanach from the geometric HahnBanach (aka first separation theorem).  HahnBanach theorem 

Oct 13  Test.  
Oct 20  Some convergence theorems; negligible sets (sets of zero measure) and functions; properties true almost everywhere.  HW7  
Oct 22  Equivalence classes of functions. Space of maps with finite seminorm N_p and related properties.  
Oct 27  pintegrable functions, L^p topology and L^p spaces. Completeness of L^p and other related spaces.  
Oct 29  Behavior of sequence and subsequences. Properties of pintegrable functions. Extension of measures and integral of functions in L^1; integrable functions.  
Nov 3  Upper envelope of functions in L^p. Monotone convergence theorem.  
Nov 5  Dominated convergence theorem. Relation between the
spaces L^p and L^q. Several remarks on integration. 

Nov 5  Upper and lower envelopes of sequences of integrable
functions. More convergence theorems. Integrability of
lower and upper semicontinuous functions. 
HW8 
Makeup class for the week of Nov 1620. Buttrick 112 at 7pm. 
Nov 10  Integrable sets and their properties. 

Nov 12  Boolean rings, Boolean algebras, and sigma algebras.
Step functions. 

Nov 12  A Riesz representation type theorem. 
Makeup class for the week of Dec 711. SC 1312 at 7pm. 

Nov 17  Student presentation: characterization of integrable sets; completion of metric spaces; extension of linear functionals.  
Dec 1  A Riesz representation type theorem.  HW9  
Dec 3  A Riesz representation type theorem.  
Dec 10  Last class  Last day to turn in homework. 