Advanced Analysis: Handwritten Notes

These notes are provided by Mr. Anwar Khan. We are really very thankful to Mr. Anwar Khan for providing these notes and appreciates his effort to publish these notes on MathsStudyForStudent.blogspot.com
It covers the complete syllabus of Advanced Analysis paper of MSc Mathematics. See the contents of the notes given below to see the topics covered by these notes.

CONTENTS OR SUMMARY:

  • Equivelent Set
  • Symmetric property
  • Transitive Property
  • Finite Set
  • Infinite Set
  • Denumerable Set
  • Countable Set
  • Geometrical Interperation of Equivalent Set
  • Duality
  • Choice Function
  • Characteristic Function
  • Cardinal Numbers
  • Arithmetic Cardinal Number
  • Transfinite Cardinal Numbers 24
  • Ordering of Cardinal Number 31
  • Exponent of Cardinal Numbers 37
  • Ordered Set 47
  • Partially Ordered Set 47
  • Reflexive 48
  • Anti-symmetric 48
  • Transitive 49
  • Inverse Ordered 55
  • First Element 57
  • Last Element 57
  • Maximal Element 60
  • Minimal Element 60
  • The Linearly Ordered Set 62
  • Lower Bound 63
  • Upper Bound 63
  • Infimum 64
  • Suprimum 64
  • Similar Sets 66
  • Order type (Ordinal Number) 74
  • Well Ordered Set 74
  • Principal of Mathematical Induction 78
  • Initial Segment 78
  • Immediate Successor & Immediate Predecessor 90
  • Shorter & Longer Sets 92
  • Ordinal Number 94
  • Transfinite Ordinal Number 95
  • Inequality and Ordinal Numbers 95
  • Ordinal Addition 95
  • Ordinal Multiplication 96
  • Structure of Ordinal Number 97
  • Ordinary Differential Equation 101
  • Hyper-geometric Series 101
  • Gamma Function 102
  • Beta Function 105
  • Relation between Gama and Beta Function 105
  • Bessel Function 128
  • Differential Recurrence relation of Bessel Function 133
  • Spherical Bessel Function 137
  • Properties of Bessel Function 137
  • Hyper-geometric Function 148
  • Legender Polynomial 159
  • Generating Function 168
  • Bessel’s differential Equation 186
  • Measure Theory 198
  • Set Function 198
  • Non-negative Function 198
  • Additivity of a Set Function 199
  • Finite Additive Set Function 199
  • Sub- additivity of a Set Function 200
  • Finitely sub Additive of a Set Function 200
  • Countable sub additive of a Set Function 200
  • Pre-measure 201
  • Outer Measure 202
  • Monotone Set Function 202
  • Scalar Multiplication of an Outer Measure 208
  • Lebegue Outer Measure of a Set of Real Number 211
  • Translation of a Set 212
  • Monotone Property 215
  • Lebesgue Measure of a set of Real Number 220
  • Boolean Algebra or Simply Algebra of Sets 223
  • Sigma Algebra 223
  • Boral Subset of Real Line 229
  • Lebesgue Measure of Measurable Set 232
  • Measurable Function 240
  • Almost Everywhere 250
  • Boral Measurable 256

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