Measure Theory Notes by Anwar Khan

Handwritten notes of measure theory by Anwar Khan. These notes are good to cover measure theory paper at master level. We are very thankful to 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

Partial contents

  • 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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