Lecture 2: Outer measures, construction of the Lebesgue measure.

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  • Опубліковано 9 лют 2025
  • Lecture 2: Section 1.3 (up until 1.3.6).
    Outer measures, construction of Lebesgue measure.

КОМЕНТАРІ • 9

  • @TheMorhaGroup
    @TheMorhaGroup 4 місяці тому +3

    "I hope you're as excited as I am😐😑😐"

  • @小小学生-r5o
    @小小学生-r5o 5 місяців тому +2

    I am still confused about what happened in 28:40. Why can we just simply add something to the left and then change the inequality the other way around?

    • @thierryrioual701
      @thierryrioual701 4 місяці тому

      Because that's the definition of the infimum:
      For all a in A, and for all epsilon > 0, a - epsilon < inf A =0, and epsilon >0, epsilon/2^k > 0.
      See "Infima and suprema of real numbers": en.wikipedia.org/wiki/Infimum_and_supremum
      Normally the inequality is strict but by taking epsilon arbitrarily close to zero the inequality becomes loose.

    • @asbjrnandersen4765
      @asbjrnandersen4765 4 місяці тому +2

      lambda star is the infimum of the RHS. thats why adding a number greater than zero flips the inequality

    • @HuggableJohn
      @HuggableJohn 2 місяці тому

      If you add a positive number to the greatest lower bound of a set, that number is no longer a lower bound of the set.

  • @darrenpeck156
    @darrenpeck156 2 роки тому

    Could this criteria lead from countable subadditivity to countable additivity?

  • @nicholaskhawli4330
    @nicholaskhawli4330 4 місяці тому

    Isnt finite union concluded from the 2 sets union proof ? I guess u could prove it but shouldnt it be trivial

  • @nicholaskhawli4330
    @nicholaskhawli4330 4 місяці тому

    Why use inequality and add the epsilon if we will just equal it to zero . Then why not just use equality .