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Lebesgue measure

Lebesgue measure

Jordan measure: finite union of element sets (boxes)
Lebesgue measure: conutable union of element sets (boxes)

Properties of Lebesgue measure

  • Monotonicity
  • Subadditivity

Simple function

  • Finite linear combination of measurable set
  • \(C_1\)\(1_{E_1}\) \(+\) \(C_2\)\(1_{E_2}\) \(...\) \(+\) \(C_k\)\(1_{E_k}\) where \(1_{E_k}\) is identity function
    Simple function

Uniform convergence

Sequence of function \(f_n\)\((x)\) is uniform convergence, if \(lim_{n\rightarrow \inf}\) \(Sup\)

\(Sup\) is the point in uniform convergence