MS-E1600 - Probability Theory D, 11.01.2021-22.02.2021
This course space end date is set to 22.02.2021 Search Courses: MS-E1600
Lecture 7: Integration theory
7.1 Stepwise approach to Lebesgue integration
Integral with respect to a general measure are first defined for finite-range functions, then extended to positive functions, and later to general measurable functions.
7.2 Definition of simple integrals
Definition of the integral for nonnegative functions with a finite range, the so-called "simple" functions.
7.3 Linearity of simple integrals
Simple integration is a linear functional.
7.4 Monotonicity of simple integrals
Simple integration is a monotone functional.
7.5 Definition of positive integrals
Extending the definition of integral to nonnegative measurable functions.
7.6 Monotonicity of positive integrals
Positive integration is a monotone functional.
7.7 Consistency lemma
Key technical consistency result about the positive integral, as formulated in [Lemma 1.18, Kallenberg 2002].
7.8 Approximating integrals by sums
Computing positive integrals in practice is done by approximating using finite sums, corresponding to simple integrals of finite-range functions pointwise approximating the integrand.
7.9 Linearity of positive integrals
Positive integration is a monotone functional.
Alternative reading material
- [Jacod & Protter, Chapter 9]
- [Williams, Chapters 5.1-5.3, 5.5-5.8, and 6.12]