14. Mon, Apr 27 [PDF] [Video]
    Bounded linear functionals on $L^p$, general case
13. Mon, Apr 20 [PDF] [Video]
    Lebesgue-Radon-Nikodym Theorem [Chap 6 Sec 4.2]
12. Fri, Apr 17 [PDF] [Video]
    spaces revisited, orthogonal projections, Riesz Representation Theorem [based on Chap 4, Sec 4 and 5.1]
11. Wed, Apr 15 [PDF] [Video]
    Signed measures, total variations, singular and absolutely continuous measures [Chap 6 Sec 4.1, 4.2 (start)]
10. Mon, Apr 13 [PDF] [Video]
    Lebesgue-Stieltjes integral and Borel measures on $\mathbb{R}$ [Chap 6 Sec 3.3]
9. Fri, Apr 10 [PDF] [Video]
    Polar (or spherical) coordinates [Chap 6 Sec 3.2]
8. Wed, Apr 8 [PDF] [Video]
    Product measures, the Fubini theorem [Chap 6 Sec 3.1]
7. Mon, Apr 6 [PDF] [Video]
    Integration on a measure space, measurable functions, simple fifunctions, Convergence Theorems, $L^p$ spaces [Chap 6 Sec 2]
6. Fri, Apr 3 [PDF] [Video]
    Premeasures and Extension Theorem [Chap 6 Sec 1.3]
5. Wed, Apr 1 [PDF] [Video]
    Metric exterior measures, Regularity of Borel measures [Chap 6 Sec 1.2]
4. Mon, Mar 30 [PDF] [Video]
    Abstract measure spaces [Chap 6 Sec 1]; Exterior measures and Caratheodory’s theorem [Chap 6 Sec 1.1]; Metric exterior measures [Chap 6 Sec 1.2 (start)]
3. Fri, Mar 27 [PDF] [Video]
    Bounded linear functionals on $L^p$; Remark on Banach spaces and their duals
2. Wed, Mar 25 [PDF] [Video]
    $L^p$ spaces for different $p$’s; $L^\infty$ space; Bounded linear functionals on $L^p$
1. Mon, Mar 23 [PDF] [Video]
    Jump functions, Differentiability of jump functions [Chap 3 Sec 3]; $L^p$ spaces, Young’s, Hölder’s, Minkowski’s inequalities; Completeness of $L^p$ (Riesz-Fischer)