# 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 |

### 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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