Measure Theory.
Measure of Plane Sets.
Systems of Sets.
Measures on Semirings. Continuation of a Measure from a Semiring to the Minimal Ring over it.
Continuations of Jordan Measures.
Countable Additivity. General Problem of Continuation of Measures.
Lebesgue Continuation of Measure, Defined on a Semiring with a Unit.
Lebesgue Continuation of Measures in the General Case.
Measurable Functions.
Definition and Basic Properties of Measurable Functions.
Sequences of Measurable Functions. Different Types of Convergence.
The Lebesgue Integral.
The Lebesgue Integral for Simple Functions.
General Definition and Basic Properties of the Lebesgue Integral.
Limiting Processes Under the Lebesgue Integral Sign.
Comparison of the Lebesgue Integral and the Riemann Integral.
Direct Products of Systems of Sets and Measures.
Expressing the Plane Measure by the Integral of a.
Linear Measure and the Geometric Definition of the Lebesgue Integral.
Fubini's Theorem.
The Integral as a Set Function.
Functions Which Are Square Integrahie.
The L2 Space.
Mean Convergence. Sets in L2 which are Everywhere Complete.
L2 Spaces with a Countable Basis.
Orthogonal Systems of Functions. Orthogonalisation.
Fourier Series on Orthogonal Systems.
Riesz-Fischer Theorem.
The Isomorphism of the Spaces L2 and l2.
The Abstract Hilbert Space. Integral Equations with a Symmetric Kernel.
Abstract Hilbert Space.
Subspaces. Orthogonal Complements. Direct Sum.
Linear and Bilinear Functionals in Hilbert Space.
Completely Continous Self-Adjoint Operators in H.
Linear Equations with Completely Continuous Operators.
Integral Equations with a Symmetric Kernel.