[Lecture Summary] Measure Theory : MA 813 of IIT Bombay, MIT 18.125

This contents is based on Lecture of MATH 813 of IIT Bombay(link) and MIT 18.125(link)

• Lecture 1 : Measurability
  • Contents
    • Prelimienary
    • $\sigma-$algebra
    • Measurable set, space, function
    • Composition property
    • Borel set
    • With sequence of function
  • Notion Link
• Lecture 2 : Abstract & Axiomatic version of Lebesgue integral
  • Contents
    • Simple function
    • Measure spae
    • Lebesgue Integral
    • Radon-Nikodym theorem
  • Notion Link
• Lecture 3 : MCT, DCT
  • Contents
    • Monotonic Convergence Theorem
    • Fatous’ Lemma
    • Absolutely Continuous Measures
    • $L^1$ space
    • Dominated Convergence theorem
  • Notion Link
• Lecture 4 : Measure zero
  • Contents
    • Almost Everywhere
    • Completion
    • Corollary of DCT
    • Avergage measure on finite space
    • Borel-Cantelli Lemma
  • Notion Link
• Lecture 5: Key concepts from Topology
  • Contents
    • Compact space, Locally compact
    • Hausdorff and Uryshon’s Lemma
    • Semi-continuous
    • Notion Link
• Lecture 6~10 : RRT Proof
• Contents
  • Preliminary Notion Link
  • Stepwise proof Notion Link