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]

Last modified: Wednesday, 3 February 2021, 3:34 PM