MA 503: Lebesgue Measure and Integration

This course begins with a review of topics normally covered in undergraduate analysis courses: open, closed and compact sets; liminf and limsup; continuity and uniform convergence. Next the course covers Lebesgue measure in Rn including the Cantor set, the concept of a sigma-algebra, the construction of a nonmeasurable set, measurable functions, semicontinuity, Egorovs and Lusin’s theorems, and convergence in measure. Next we cover Lebesgue integration, integral convergence theorems (monotone and dominated), Tchebyshev’s inequality and Tonelli’s and Fubini’s theorems. Finally Lp spaces are introduced with emphasis on L2 as a Hilbert space. Other related topics will be covered at the instructor’s discretion.