8.1 Signed integrals

Lebesgue integral of a signed function is defined by separately integrating its positive and negative parts.

8.2 Where is almost everywhere?

Values outside a set of measure zero determine the integral of a function.

8.3 Lebesgue vs Riemann

Lebesgue integral against a Lebesgue measure coincides with the classical Riemann integral for Riemann-integrable Borel functions.

8.4 Expectations

Lebesgue integrals against a probability measure are expectations of random variables.

8.5 Integrating against the law

The expectation of a random variable can be computed by integrating against the law of the random variable.

8.6 Integrating against a density

Laws of some random variables admit a probability density function with respect to the Lebesgue measure on the real line. Expectations related to such random variables reduce to familiar formulas (e.g. normal distribution, exponential distribution).

8.7 Monotone and dominated continuity

Integration as a linear functional is not continuous in general, but continuity holds for monotone and suitably dominated function sequences.


Alternative reading material

  • [Jacod & Protter, Chapter 9]
  • [Williams, Chapters A5, 5.4, 5.9, 6.1, 6.2, 6.5]


Last modified: Wednesday, 3 February 2021, 10:04 PM