Analysis
Section outline
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Sigma algebras: definition and examples. Abstract measures: definition and examples.
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Counting measure. Properties of abstract measures: monotonicity, finite additivity and measure of the union/difference, continuity from below and from above, examples.
lebesgue measure: introduction, intervals, definition of outer measure.
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Exercises on abstract measures. Countable sub-additivity. Lebesgue outer measure, main properties.
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Lebesgue class, basic facts and properties. Measurable functions: definition and examples (indicator and simple functions).
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