Videos
Section outline
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Definition of \(\sigma-\)algebra of sets. Examples. \(\sigma-\)algebra generated by a family of sets. Definition of abstract measure. Dirac measure.
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Counting measure. Basic properties of abstract measures. Continuity properties. Definition of Lebesgue's outer measure.
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Exercises on abstract measures. Properties of Lebesgue's outer measure, Vitali's example of non countable additivity.
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Lebesgue class. Basic properties. Measurable functions: definition and first examples.
The video recording has been interrupted by a blackout.
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Continuous functions on \(\Bbb R^k\) are Lebesgue measurable. Complete measures. Algebraic composition of measurable functions. Chain rule. Point-wise limit of measurable functions.
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Exercises on Lebesgue's measure. Lebesgue's definition of abstract integral.
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Pointwise approximation of positive measurable functions through simple functions. Definition of integral for real and complex valued functions, and main properties. Chebyshev inequality.
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Lebesgue's integral on \(\Bbb R^k\) and connections with Riemann and generalized integrals. Examples. Fubini's theorem.
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Exercises on abstract integrals. Beppo-Levi's and monotone convergence theorems. Examples.
Unfortunately the recording suffered of technical problems and interrupts after 1/3 of the class.
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Monotone convergence for decreasing sequences. Examples. Lebesgue's dominated convergence theorem. Exercise.
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Exercises on dominated convergence. Integrals depending on parameters, continuity and differentiability theorems.
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Fourier transform of the Gaussian distribution. Exercises on differentiaiblity theorem.
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Stronger and equivalent norms. Finite dimensional norms are always equivalent. Examples with infinite dimensional norms. \( L^p\) spaces.
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Exercises on normed spaces. L^2 norm: Cauchy-Schwarz inequality. Essentially bounded functions, \( L^\infty \) norm.
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Limits in normed spaces: definition, examples, Cauchy property. Uniform convergence.
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Exercises on convergence in norm. Completeness of \(B(X)\) and \(\mathscr C(K)\), \(K\subset\Bbb R^d\) compact. Continuity of uniform limit. Convergence in \(L^p\) norm.
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Exercises on convergence. Convolution: definition, Young's inequality, example.
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Approximate units, mollification theorem. Scalar and hermitian products. Examples. Canonical norm of an inner product space, abstract Cauchy-Schwarz inequality. Orthogonal vectors, Pythagorean theorem, parallelogram identity. Exercises.
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Orthogonal projection: existence, uniqueness and characterization. Examples and exercises.
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Orthogonal complement: exercises. Series of vectors: Weierstrass' test and convergence of orthogonal series. Orthonormal bases.
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Orthonormal basis for \( L^2([a,b]) \) , classical Fourier series. \( L^1\) Fourier Transform: definition, remarks and first examples.
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FT of rectangle. Basic properties of FT: continuity, boundedness, Riemann-Lebesgue lemma. FT of derivative and derivative of FT. FT maps Schwarz functions space into itself. FT of convolution product. Exercises.
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Exercise. Inversion formula. FT of Cauchy distribution. Exercise.
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Injectivity of FT, Fourier original and inverse FT. Exercises
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