Measure and Integration, Lectures

Core content

• Preliminaries: unions and intersections with general index sets, countable and uncountable sets, the limit superior and the limit inferior, the Riemann integral.
• Sigma algebra and measure. Outer measure and measurable sets, Borel sets.
• Lebesgue outer measure, Lebesgue measure, and Lebesgue measurable sets.
  
• Measurable functions.
• The integral with respect to a measure: the integral of a simple function, the integral of a nonnegative function, the Monotone Convergence Theorem, the integral of a measurable function, and the Dominated Convergence Theorem.
• The relationship between the Riemann and Lebesgue integrals.
  

Complementary knowledge

• An example of a non-Lebesgue-measurable set.
• The Lebesgue integral in Rⁿ and Fubini's Theorem.
  
• The Axiom of Choice.
• L^p spaces.

Alternative learning events among themselves

Group 1: Exercise session

10.01.2022 09:00 - 12:00, TAU Tietotalo TD308 Harjoitussali (20)
17.01.2022 09:00 - 12:00, TAU Tietotalo TD308 Harjoitussali (20)
24.01.2022 09:00 - 12:00, TAU Tietotalo TD308 Harjoitussali (20)
31.01.2022 09:00 - 12:00, TAU Tietotalo TD308 Harjoitussali (20)
07.02.2022 09:00 - 12:00, TAU Tietotalo TD308 Harjoitussali (20)
14.02.2022 09:00 - 12:00, TAU Tietotalo TD308 Harjoitussali (20)
21.02.2022 09:00 - 12:00, TAU Tietotalo TD308 Harjoitussali (20)

Common learning events for all

Lecture: Lecture

11.01.2022 12:00 - 14:00, TAU Rakennustalo RI207 harjoitussali (70)
12.01.2022 08:00 - 10:00, TAU Rakennustalo RI207 harjoitussali (70)
18.01.2022 12:00 - 14:00, TAU Rakennustalo RI207 harjoitussali (70)
19.01.2022 08:00 - 10:00, TAU Rakennustalo RI207 harjoitussali (70)
25.01.2022 12:00 - 14:00, TAU Rakennustalo RI207 harjoitussali (70)
26.01.2022 08:00 - 10:00, TAU Rakennustalo RI207 harjoitussali (70)
01.02.2022 12:00 - 14:00, TAU Rakennustalo RI207 harjoitussali (70)
02.02.2022 08:00 - 10:00, TAU Rakennustalo RI207 harjoitussali (70)
08.02.2022 12:00 - 14:00, TAU Rakennustalo RI207 harjoitussali (70)
09.02.2022 08:00 - 10:00, TAU Rakennustalo RI207 harjoitussali (70)
15.02.2022 12:00 - 14:00, TAU Rakennustalo RI207 harjoitussali (70)
16.02.2022 08:00 - 10:00, TAU Rakennustalo RI207 harjoitussali (70)
22.02.2022 12:00 - 14:00, TAU Rakennustalo RI207 harjoitussali (70)
23.02.2022 08:00 - 10:00, TAU Rakennustalo RI207 harjoitussali (70)

Compulsory Prerequisites

• Mathematical Analysis, MAT-60200, 5 cr
• Multivariable Calculus, MATH.APP.220, 5 cr
• Multivariable Calculus, MATH.APP.220, 5 cr
• Differential and integral calculus of several variables, MATH.MA.630, 5 cr
• Euclidean Spaces, MATH.MA.620, 5 cr
• Calculus of several variables, MTTMA4, 5 cr
• Calculus of several variables, MTTMA5, 5 cr

• Axler, Sheldon: Measure, Integration & Real Analysis. Published with a Creative Commons Attribution-NonCommercial license, see https://measure.axler.net/

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