Measure Theory Notes by Anwar Khan

Handwritten notes of measure theory by [[people:anwar-khan]]. These notes are good to cover measure theory paper at master level. We are very thankful to [[people:anwar-khan]] for sending these notes.

Name Measure Theory: Notes
Provider Mr. Anwar Khan
Pages 169 pages
Format PDF (see Software section for PDF Reader)
Size 7.75 MB
  • Algebra on $X$
  • Sigma Algebra i.e. $\sigma-$algebra on $X$
  • Trivial $\sigma-$algebra; Largest $\sigma-$algebra
  • Increasin & sequence of sets
  • Decreasing sequence of sets
  • Define $\lim\limits_{k\to \infty} \sup A_k$ and $\lim\limits_{k\to \infty} \inf A_k$
  • Smallest $\sigma-$algebra
  • Borel set & Borel $\sigma-$algebra
  • $G_\sigma-$set; $F_\sigma-$set
  • Set of extended real numbers; Set function; Properties of set function
  • Measure
  • Finite measure; $\sigma-$finite measure
  • Monotone convergence theorem
  • Measurable space and measure space; Finite measure space; $\sigma-$finite measure space; $\mathcal{A}-$measurable set
  • $\sigma-$finite set
  • Null set
  • Complete $\sigma-$algebra; Complete
  • measure space; Outer measure
  • $\mu^*-$measurable set
  • Lebesgue outer measure
  • Lebesgue measurable set or $\mu^*-$measurable set; Lebesgue $\sigma-$algebra; Lebesgue measurable space
  • Lebesgue measure space
  • Dense sub set of $X$
  • Translation of a set; Dielation of a set
  • Translation invarient
  • Addition modulo 1
  • Translation of $E$ mod 1
  • Measurable function
  • Characteristic function
  • Almost every where property; Equal almost every where
  • Limit inferior and limit superior of real value sequence
  • Sequence of $\mathcal{A}-$measurable functions & its limits & their properties
  • Larger & smaller of two function; Positive part of $f$; Negative part of $f$; Absolute function of $f$
  • Limit existence almost every where
  • Step function
  • Riemann integral
  • Simple function; Canonical representation of simple function
  • Lebesgue integral of simple function
  • Bounded function; Lower Lebesgue integral; Upper Lebesgue integral
  • Lebesgue integral of bounded function
  • Uniform convergence
  • Almost uniform convergence; Egoroff's theorem; Bounded convergence theorem; Non-negative function; Lebesgue integralof non-negative function
  • Monotone convergence theorem
  • Fatou's lemma

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