Contents
Tutorial
1 :Dynkin systems
Dynkin system ,
Sigma-algebra ,
Dynkin theorem
Tutorial 2 :Caratheodory
Measure , Outer-measure ,
Extension of measures
Tutorial 3 :Stieltjes measure
Stieltjes
measure , Lebesgue
measure ,
Borel sigma-algebra
Tutorial 4 :Measurability
Continuous map , Measurable map ,
Metric Topology
Tutorial 5 :Lebesgue integral
Monotone convergence ,
Fatou lemma ,
Dominated
convergence
Tutorial 6 :Product spaces
Rectangle , Product sigma-algebra ,
Product topology
Tutorial
7 :Fubini theorem
Product measure , Partial measurability ,
Fubini theorem
Tutorial
8 :Jensen inequality
Convex function , Compact space ,
Taylor expansion , Jensen inequality
Tutorial
9 :Lp - spaces
Holder
inequality , Cauchy-Schwarz , Minkowski , Lp-Completeness
Tutorial 10 :L 2 -functionals
Complete spaces , Hilbert spaces ,
Orthog. projection ,
L2-Functionals
Tutorial 11 :Complex measure
Complex measure , Signed measure ,
Total variation of a
measure
Tutorial 12 :Radon-Nikodym
Absolute continuity ,
Radon-Nikodym ,
Hahn decomposition
Tutorial 13 :Regular measure
Inner, Outer-regular measure ,
Local compactness , Compact support
Tutorial 14 : Finite variation
Maps of finite, bounded
variation ,
Complex stieltjes
measure
Tutorial 15 :Stieltjes integral
Absolutely
continuous maps ,
Stieltjes
integral , Change of time
Tutorial 16 :Differentiation
Fundamental
calculus theorem ,
Lebesgue point , Connected space
Tutorial
17 :Image measure
Distribution , Integral projection th.
Linear transf. of
Lebesgue measure
Tutorial 18 :Jacobian formula
Differentiable map , Partial derivatives ,
Jacobian , Jacobian formula
Tutorial 19 :Fourier transform
Convolution , Dirac distribution ,
Fourier transform , Narrow convergence
Tutorial 20 :Gaussian measure
Normal distribution ,
Gaussian vector ,
Covariance , Normal density
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Site created on
15-May-99